http://wiki.manchester.ac.uk/blocks/api.php?action=feedcontributions&feedformat=atom&user=CesareGArditoBlock library - User contributions [en-gb]2026-06-25T09:25:33ZUser contributionsMediaWiki 1.30.1http://wiki.manchester.ac.uk/blocks/index.php?title=User:CesareGArdito&diff=1250User:CesareGArdito2025-12-19T20:19:00Z<p>CesareGArdito: /* Cesare Giulio Ardito */</p>
<hr />
<div>[[Image:CesareG_Ardito.jpg|150px|thumb|right]]<br />
<br />
== Cesare Giulio Ardito ==<br />
<br />
I am currently a lecturer in mathematics at the University of Manchester. I am originally from Rome, Italy, where I got my Bachelor’s and Master’s Degree at Sapienza University of Rome.<br />
<br />
My general research interests lie in finite group theory, in particular modular representation theory and global-local conjectures. <br />
<br />
[https://cesaregardito.wordpress.com/ My homepage]<br />
<br />
[http://orcid.org/0000-0002-0989-8609 My ORCiD]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Tame_blocks&diff=1208Tame blocks2022-11-09T18:28:10Z<p>CesareGArdito: </p>
<hr />
<div>A finite dimensional <math>k</math>-algebra <math>A</math> is said to have ''finite representation type'' if there are only finitely many isomorphism classes of indecomposable modules. Algebras of infinite representation type are split into two cases: ''tame'' and ''wild''. For definitions see Section I.4 of [[References#E|[Er90]]], but tame essentially means that almost all modules of a given dimension fit into finitely many one-parameter families and wild means that the module category is comparable to that for <math>k\langle X,Y \rangle</math>. The properties of having finite or infinite representation type and of being tame or wild are all Morita invariants.<br />
<br />
Let <math>B</math> be a block of <math>kG</math> for a finite group <math>G</math> with defect group <math>D</math>. Then<br />
*<math>B</math> has finite representation type if and only if <math>D</math> is cyclic.<br />
*<math>B</math> is tame if and only if <math>D</math> contains no noncyclic abelian <math>p</math>-subgroup of order greater than four. Equivalently, <math>D</math> is generalized quaternion, dihedral or semidihedral.<br />
*Otherwise, <math>B</math> is wild.<br />
<br />
In a series of papers and her book [[References#E|[Er90]]], Erdmann describes the basic algebra of tame type (see page vi of [[References#E|[Er90]]] for a definition), and in most cases describes which of these occur as basic algebras for blocks of finite groups. As a consequence the <math>k</math>-Donovan conjecture holds for all tame blocks except when <math>D</math> is generalized quaternion of order greater than eight and <math>l(B)=2</math>. In this last case there is an infinite class of basic algebras where it is not determined which may occur as basic algbras for blocks of finite groups. In general it is difficult to extend these results to blocks defined over <math>\mathcal{O}</math>. For blocks with Klein four defect groups this was done by Linckelmann in [[References#L|[Li94]]], and for blocks with generalized quaternion defect groups and <math>l(B)=3</math> by Eisele in [[References#E|[Ei16]]]. Note that by Erdmann's classifications tame blocks with just one simple module are nilpotent, and so the classification extends to <math>\mathcal{O}</math> in this case. <br />
<br />
Linckelmann's results on Klein four defect groups were extended using the classification of finite simple groups to classify such blocks up to Puig equivalence in [[References#C|[CEKL11]]].<br />
<br />
[[Statements of conjectures|Puig's conjecture]] holds for all principal tame blocks by [[References#K|[KoLa20]]], [[References#K|[KoLa20b]]], [[References#K|[KoLaSa22]]] .<br />
<br />
<!--== Notes ==<br />
<references /> <br />
--></div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Tame_blocks&diff=1207Tame blocks2022-11-09T18:27:15Z<p>CesareGArdito: </p>
<hr />
<div>A finite dimensional <math>k</math>-algebra <math>A</math> is said to have ''finite representation type'' if there are only finitely many isomorphism classes of indecomposable modules. Algebras of infinite representation type are split into two cases: ''tame'' and ''wild''. For definitions see Section I.4 of [[References#E|[Er90]]], but tame essentially means that almost all modules of a given dimension fit into finitely many one-parameter families and wild means that the module category is comparable to that for <math>k\langle X,Y \rangle</math>. The properties of having finite or infinite representation type and of being tame or wild are all Morita invariants.<br />
<br />
Let <math>B</math> be a block of <math>kG</math> for a finite group <math>G</math> with defect group <math>D</math>. Then<br />
*<math>B</math> has finite representation type if and only if <math>D</math> is cyclic.<br />
*<math>B</math> is tame if and only if <math>D</math> contains no noncyclic abelian <math>p</math>-subgroup of order greater than four. Equivalently, <math>D</math> is generalized quaternion, dihedral or semidihedral.<br />
*Otherwise, <math>B</math> is wild.<br />
<br />
In a series of papers and her book [[References#E|[Er90]]], Erdmann describes the basic algebra of tame type (see page vi of [[References#E|[Er90]]] for a definition), and in most cases describes which of these occur as basic algebras for blocks of finite groups. As a consequence the <math>k</math>-Donovan conjecture holds for all tame blocks except when <math>D</math> is generalized quaternion of order greater than eight and <math>l(B)=2</math>. In this last case there is an infinite class of basic algebras where it is not determined which may occur as basic algbras for blocks of finite groups. In general it is difficult to extend these results to blocks defined over <math>\mathcal{O}</math>. For blocks with Klein four defect groups this was done by Linckelmann in [[References#L|[Li94]]], and for blocks with generalized quaternion defect groups and <math>l(B)=3</math> by Eisele in [[References#E|[Ei16]]]. Note that by Erdmann's classifications tame blocks with just one simple module are nilpotent, and so the classification extends to <math>\mathcal{O}</math> in this case. <br />
<br />
Linckelmann's results on Klein four defect groups were extended using the classification of finite simple groups to classify such blocks up to Puig equivalence in [[References#C|[CEKL11]]].<br />
<br />
Puig's conjecture holds for all principal tame blocks by [[References#K|[KoLa20]]], [[References#K|[KoLa20b]]], [[References#K|[KoLaSa22]]] .<br />
<br />
<!--== Notes ==<br />
<references /> <br />
--></div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Status_of_Donovan%27s_conjecture&diff=1206Status of Donovan's conjecture2022-11-09T18:25:12Z<p>CesareGArdito: /* Donovan's conjecture by p-group */</p>
<hr />
<div>[[Image:Donovan.jpg|150px|thumb|right|Peter Donovan]]<br />
<br />
== Donovan's conjecture by <math>p</math>-group ==<br />
<br />
In the following, the column headed [[Statements of conjectures #Donovan's conjecture|Donovan's conjecture]] indicates whether the conjecture is known over <math>k</math> or <math>\mathcal{O}</math>.<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| <math>p</math>-groups<br />
! scope="col"| Donovan's conjecture<br />
! scope="col"| Puig's conjecture<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|-<br />
|Cyclic <math>p</math>-groups || <math>\mathcal{O}</math> || Yes || [[References#L|[Li96]]] ||<br />
|-<br />
|<math>C_2 \times C_2</math> || <math>\mathcal{O}</math> || Yes || [[References#C|[CEKL11]]] || Donovan's conjecture without CFSG, Puig using CFSG<br />
|-<br />
|Abelian <math>2</math>-groups || <math>\mathcal{O}</math> || No || [[References#E|[EEL18]]] ||<br />
|-<br />
|Abelian <math>3</math>-groups || No || No || [[References#K|[Ko03]]] || Puig's conjecture known for principal blocks<br />
|-<br />
|Dihedral <math>2</math>-groups || <math>k</math> || No || [[References#E|[Er87]]] ||<br />
|-<br />
|Semidihedral <math>2</math>-groups || <math>k</math> || No || [[References#E|[Er88c], [Er90b]]] ||<br />
|-<br />
|<math>Q_8</math> || <math>\mathcal{O}</math> || No || [[References#E|[Er88a]]], [[References#E|[Er88b]]], [[References#K|[HKL07]]], [[References#E|[Ei16]]] ||<br />
|-<br />
|<math>Q_8 \times C_{2^n}</math> || <math>\mathcal{O}</math> || No || [[References#E|[EL20]]] ||<br />
|-<br />
|<math>Q_8 \times Q_8</math> || <math>\mathcal{O}</math> || No || [[References#E|[EL20]]] ||<br />
|-<br />
|Generalised quaternion <math>2</math>-groups || No || No || [[References#E|[Er88a], [Er88b]]] || Donovan's conjecture over <math>\mathcal{O}</math> known if <math>l(B) \neq 2</math><ref>When <math>l(B) \neq 2</math>, each <math>k</math>-Morita equivalence class lifts uniquely to <math>\mathcal{O}</math> by [[References|[Ei16]]].</ref><br />
|- <br />
|Minimal nonabelian <math>2</math>-groups of the form <math>\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> || <math>\mathcal{O}</math> || No || [[References#E|[EKS12]]] || <br />
|-<br />
|Metacyclic noncyclic <math>2</math>-groups of nonmaximal class || <math>\mathcal{O}</math> || No || [[References#C|[CG12]]], [[References#S|[Sa12b]]] || All blocks nilpotent<br />
|}<br />
<br />
== Donovan's conjecture by class of group or block ==<br />
<br />
In the table, the column headed Donovan's conjecture indicates whether the conjecture is known over <math>k</math> or <math>\mathcal{O}</math>.<br />
<br />
Note that knowing the <math>\mathcal{O}</math>-Donovan conjecture or [[Statements of conjectures #Puig's conjecture|Puig's conjecture]] for blocks for a class of groups does not necessarily mean that the <math>\mathcal{O}</math>-lifts or source algebras of the <math>k</math>-Morita equivalence classes involved are known. This is only known for elements of the Morita equivalence class which occur as blocks of groups in that class.<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Groups<br />
! scope="col"| Blocks<br />
! scope="col"| Donovan's conjecture<br />
! scope="col"| Puig's conjecture<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|-<br />
|<math>p</math>-solvable groups || All || <math>\mathcal{O}</math> || Yes || Over <math>k</math> by [[References|[Ku81]]], Puig's conjecture by [[References|[Pu09]]] || See [[References#L|[Li18d,10.6.2]]]<br />
|-<br />
|Symmetric groups || All || <math>\mathcal{O}</math> || Yes || Over <math>k</math> by [[References#S|[Sc91]]], Puig's conjecture by [[References#P|[Pu94]]] ||<br />
|-<br />
|Double covers of symmetric groups || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke96]]] ||<br />
|-<br />
|Alternating groups and their double covers || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke02], [Ke96]]] ||<br />
|-<br />
|<math>GL_n(q)</math> for fixed <math>q</math> || Unipotent blocks || <math>\mathcal{O}</math> || Yes || Over <math>k</math> by [[References#J|[Jo96]]], Puig's conjecture by [[References#K|[Ke01]]] ||<br />
|-<br />
|Classical groups || Unipotent blocks for linear primes || <math>\mathcal{O}</math> || Yes || [[References#H|[HK00], [HK05]]] || Detailed results beyond those stated here<br />
|-<br />
|Weyl groups of type <math>B, D</math> || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke00]]] ||<br />
|-<br />
|Arbitrary groups || Blocks with [[Glossary#Trivial intersection subgroup|trivial intersection]] defect groups || <math>\mathcal{O}</math> || No || [[References#A|[AE04]]] ||<br />
|}<br />
<br />
== Weak Donovan conjecture ==<br />
<br />
As described in [[References #D|[Dü04]]] the [[Statements of conjectures #Weak Donovans conjecture|Weak Donovan conjecture]] is equivalent to [[Statements of conjectures #Weak Donovans conjecture|bounding the dimensions of the Ext spaces between simple modules]] and [[Statements of conjectures #Weak Donovans conjecture|bounding the Loewy length]]. See [[References #G|[GT19]]] and [[References #S|[Sh20]]] for progress on the former problem.<br />
<br />
== Notes ==<br />
<br />
<references /></div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1205References2022-11-09T18:24:00Z<p>CesareGArdito: </p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
|- id="A"<br />
|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.<br />
|-<br />
|[Al80] || '''J. L. Alperin''', ''Local representation theory'', The Santa Cruz Conference on Finite Groups., Proc. Sympos. Pure Math. '''37''' (1980), 369-375.<br />
|-<br />
|[AE81] || '''J. L. Alperin and L. Evens''', ''Representations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71–78. <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.<br />
|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.<br />
|-<br />
|[BP80] || '''M. Broué and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.<br />
|-<br />
|[BP80b] || '''M. Broué and L. Puig''', ''Characters and local structure in G-algebras'', J. Algebra '''63''' (1980), 306-317.<br />
|- id="C"<br />
|[Cr11] || '''D. A. Craven''', ''The Theory of Fusion Systems: An Algebraic Approach'', Cambridge University Press (2011).<br />
|-<br />
|[Cr12] || '''D. A. Craven''', [https://arxiv.org/abs/1207.0116 ''Perverse Equivalences and Broué's Conjecture II: The Cyclic Case''], [https://arxiv.org/abs/1207.0116 arXiv:1207.0116]<br />
|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Broué's conjecture'', Adv. Math. '''248''' (2013), 1-58.<br />
|-<br />
|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
|-<br />
|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
|- id="D"<br />
|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
|[Ei19] || '''F. Eisele''', ''[https://arxiv.org/abs/1908.00129 On the geometry of lattices and finiteness of Picard groups]'', [https://arxiv.org/abs/1908.00129 arXiv:1908.00129]<br />
|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', [https://arxiv.org/abs/2006.13837 arXiv:2006.13837]<br />
|-<br />
|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
|-<br />
|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.<br />
|-<br />
|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.<br />
|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).<br />
|-<br />
|[Er90b] || '''K. Erdmann''', ''Algebras and semidihedral defect groups II'', Proc. LMS '''60''' (1990), 123-165.<br />
|- id="F"<br />
|[Fa17] || '''N. Farrell''', ''On the Morita Frobenius numbers of blocks of finite reductive groups'', J. Algebra '''471''' (2017), 299-318.<br />
|-<br />
|[FK18] || '''N. Farrell and R. Kessar''', [https://arxiv.org/abs/1805.02015 ''Rationality of blocks of quasi-simple finite groups''], Represent. Theory '''23''' (2019), 325-349.<br />
|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], [https://arxiv.org/abs/2016.07231 arXiv:2016.07231]<br />
|-<br />
|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).<br />
|-<br />
|[GT19] || '''R. M. Guralnick and Pham Huu Tiep''', ''Sectional rank and Cohomology'', J. Algebra (2019) https://doi.org/10.1016/j.jalgebra.2019.04.023<br />
|- id="H"<br />
|[HM07] || '''G. T. Helleloid and U. Martin''', ''The automorphism group of a finite <math>p</math>-group is almost always a <math>p</math>-group'', J. Algebra (2007), 294-329.<br />
|-<br />
|[HP94] || '''H-W. Henn and S. Priddy''', ''<math>p</math>-nilpotence, classifying space indecompsability, and other properties of almost finite groups'', Comment. Math. Helvetici (1994), 335-350.<br />
|- <br />
|[HK00] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups'', J. Algebra '''230''' (2000), 378-423.<br />
|-<br />
|[HK05] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups II'', J. Algebra '''283''' (2005), 522-563.<br />
|-<br />
|[Ho97] || '''T. Holm''', ''Derived equivalent tame blocks'', J. Algebra '''194''' (1997), 178-200.<br />
|-<br />
|[HKL07] || '''T. Holm, R. Kessar and M. Linckelmann''', ''Blocks with a quaternion defect group over a 2-adic ring: the case <math>\tilde{A}_4</math>'', Glasgow Math. J. '''49''' (2007), 29–43.<br />
|- id="J"<br />
|[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.<br />
|-<br />
|[Jo96] || '''T. Jost''', ''Morita equivalences for blocks of finite general linear groups'', Manuscripta Math. '''91''' (1996), 121-144.<br />
|- id="K"<br />
|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.<br />
|-<br />
|[Ke00] || '''R. Kessar''', ''Equivalences for blocks of the Weyl groups'', Proc. Amer. Math. Soc. '''128''' (2000), 337-346.<br />
|-<br />
|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. <br />
|-<br />
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|-<br />
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|-<br />
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|-<br />
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|- <br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|}</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1204References2022-11-09T18:22:28Z<p>CesareGArdito: </p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
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|-<br />
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|-<br />
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|-<br />
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|- id="J"<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="L"<br />
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|- <br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|}</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1203References2022-11-09T18:20:58Z<p>CesareGArdito: </p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
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|- id="G"<br />
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|- <br />
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|- id="J"<br />
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|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.<br />
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|-<br />
|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. <br />
|-<br />
|[Ke02] || '''R. Kessar''', ''Scopes reduction for blocks of finite alternating groups'', Quart. J. Math. '''53''' (2002), 443-454.<br />
|-<br />
|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.<br />
|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
|[Ki84] || '''M. Kiyota''', ''On 3-blocks with an elementary abelian defect group of order 9'', J. Fac. Sci. Univ. Tokyo Sect. IA Math. '''31''' (1984), 33–58.<br />
|-<br />
|[Ko03] || '''S. Koshitani''', ''Conjectures of Donovan and Puig for principal <math>3</math>-blocks with abelian defect groups'', Comm. Alg. '''31''' (2003), 2229-2243; ''Corrigendum'', '''32''' (2004), 391-393.<br />
|-<br />
|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. <br />
|-<br />
|[KKW04] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's abelian defect group conjecture for Held group and the sporadic Suzuki group'', J. Algebra '''279''' (2004), 638-666. <br />
|-<br />
|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.<br />
|-<br />
|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.<br />
|-<br />
|[Kü80] || '''B. Külshammer''', ''On 2-blocks with wreathed defect groups'', J. Algebra '''64''' (1980), 529–555.<br />
|-<br />
|[Kü81] || '''B. Külshammer''', ''On p-blocks of p-solvable groups'', Comm. Alg. '''9''' (1981), 1763-1785.<br />
|-<br />
|[Kü91] || '''B. Külshammer''', ''Group-theoretical descriptions of ring-theoretical invariants of group algebras'', in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progr. Math. '''95''', pp. 425-442, Birkhauser (1991).<br />
|-<br />
|[Kü95] || '''B. Külshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.<br />
|-<br />
|[KS13] || '''B. Külshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.<br />
|-<br />
|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.<br />
|-<br />
|[Kup69] || '''H. Kupisch''', ''Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe'', Math. Z. '''108''' (1969), 77-104.<br />
|- id="L"<br />
|[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.<br />
|-<br />
|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.<br />
|-<br />
|[Li94b] || '''M. Linckelmann''', ''A derived equivalence for blocks with dihedral defect groups'', J. Algebra '''164''' (1994), 244-255. <br />
|-<br />
|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.<br />
|-<br />
|[Li18] || '''M. Linckelmann''', [https://arxiv.org/abs/1805.08884 ''The strong Frobenius numbers for cyclic defect blocks are equal to one''], [https://arxiv.org/abs/1805.08884 arXiv:1805.08884]<br />
|-<br />
|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of finite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.<br />
|-<br />
|[Li18c] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 1'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[LM20] || '''M. Linckelmann and W. Murphy''', [https://arxiv.org/abs/2005.02223 ''A 9-dimensional algebra which is not a block of a finite group''], [https://arxiv.org/abs/2005.02223 arXiv:2005.02223]<br />
|-<br />
|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.<br />
|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
|-<br />
|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)<br />
|-<br />
|[Ma86] || '''U. Martin''', ''Almost all <math>p</math>-groups have automorphism group a <math>p</math>-group'', Bull. AMS '''15''' (1986), 78-82.<br />
|-<br />
|[McK19] || '''E. McKernon''', [https://arxiv.org/abs/1912.03222 ''2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle''], J. Algebra '''563''' (2020), 30–48.<br />
|-<br />
|[MS08] || '''J. Müller and M. Schaps''', ''The Broué conjecture for the faithful 3-blocks of <math>4.M_{22}</math>'', J. Algebra '''319''' (2008), 3588-3602.<br />
|- id="N"<br />
|[NS18] || '''G. Navarro and B. Sambale''', ''On the blockwise modular isomorphism problem'', Manuscripta Math. '''157''' (2018), 263-278.<br />
|- <br />
|[Ne02] || '''G. Nebe''', [http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/survey.pdf ''Group rings of finite groups over p-adic integers, some examples''], Proceedings of the conference Around Group rings (Edmonton) Resenhas '''5''' (2002), 329-350.<br />
|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
|[Pu99]|| '''L. Puig''', ''On the local structure of Morita and Rickard equivalences between Brauer blocks'', Progress in Math. '''178''', Birkhauser Verlag (1999).<br />
|-<br />
|[Pu09] || '''L. Puig''', ''Block source algebras in p-solvable groups'', Michigan Math. J. '''58''' (2009), 323-338.<br />
|- id="R"<br />
|[Ri96] || '''J. Rickard''', ''Splendid equivalences: derived categories and permutation modules'', Proc. London Math. Soc. '''72''' (1996), 331-358.<br />
|-<br />
|[Ro95] || '''R. Rouquier''', ''From stable equivalences to Rickard equivalences for blocks with cyclic defect'', Proceedings of Groups 1993, Galway-St. Andrews Conference, Vol. 2, London Math. Soc. Lecture Note Ser. '''212''', Cambridge University Press (1995), 512-523.<br />
|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
|[Sa11] || '''B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups'', J. Algebra '''337''' (2011), 261–284.<br />
|-<br />
|[Sa12] || '''B. Sambale''', ''Blocks with defect group <math>D_{2^n} \times C_{2^m}</math>'', J. Pure Appl. Algebra '''216''' (2012), 119–125.<br />
|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
|[Sh20] || '''V. Shalotenko''', ''Bounds on the dimension of Ext for finite groups'', J. Algebra '''550''' (2020), 266-289.<br />
|-<br />
|[St02] || '''R. Stancu''', ''Almost all generalized extraspecial p-groups are resistant'', J. Algebra '''249''' (2002), 120-126.<br />
|-<br />
|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="T"<br />
|[Th93] || '''J. Thévenaz''', ''Most finite groups are <math>p</math>-nilpotent'', Exposition. Math. '''11''' (1993), 359-363.<br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1162Generic classifications by p-group class2021-01-13T20:22:14Z<p>CesareGArdito: Added section about cyclic inertial quotient acting freely</p>
<hr />
<div>This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes. <br />
<br />
== Fusion trivial ''p''-groups ==<br />
<br />
''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.<br />
<br />
Blocks with fusion trivial defect groups must be nilpotent by [[References#P|[Pu88]]].<br />
<br />
Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).<br />
<br />
Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups. <br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most three ==<br />
<br />
These have been classified in [[References#W|[WZZ18]]] and [[References#E|[EL18a]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math><br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 3</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|}<br />
<br />
<!--== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.--><br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.<br />
<br />
<br />
== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group == <br />
<br />
<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E|[ArMcK20]]].<br />
<br />
If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>. <br />
<br />
If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>. <br />
<br />
In each case, the Morita equivalence between the block and the class representative is known to be basic.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=(C2)%5E5&diff=1155(C2)^52020-11-13T09:40:21Z<p>CesareGArdito: /* Blocks with defect group (C_2)^5 */</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>(C_2)^5</math> ==<br />
<br />
These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]].<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]<br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(32,51,1)]] || <math>k((C_2)^5)</math> || 1 ||32 ||1 ||<math>1</math> || <math>(C_2)^5:GL_5(2)</math> || ||1 ||1 ||<br />
|-<br />
|[[M(32,51,2)]] || <math>k(A_4 \times (C_2)^3)</math> || 1 ||32 ||3 ||<math>C_3</math> || <math>((C_2)^3 : GL_3(2)) \times S_3</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,3)]] || <math>B_0(k(A_5 \times (C_2)^3))</math> || 1 ||32 ||3 ||<math>C_3</math> || <math>((C_2)^3 : GL_3(2)) \times C_2</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,4)]] || <math>k(((C_2)^4 :C_3) \times C_2)</math> || 1 ||16 ||3 ||<math>C_3</math> || || ||1 ||1 || The action of <math>C_3</math> comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math><br />
|-<br />
|[[M(32,51,5)]] || <math>k(((C_2)^4 :C_5) \times C_2)</math> || 1 ||16 ||5 ||<math>C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,6)]] || <math>k(((C_2)^3:C_7) \times (C_2)^2)</math> || 1 ||32 ||7 ||<math>C_7</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,7)]] || <math>B_0(k(SL_2(8) \times (C_2)^2))</math> || 1 ||32 ||7 ||<math>C_7</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,8)]] || <math>k(A_4 \times A_4 \times C_2)</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,9)]] || <math>B_0(k(A_4 \times A_5 \times C_2))</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||<br />
|-<br />
|[[M(32,51,10)]] || <math>B_0(k(A_5 \times A_5 \times C_2))</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,11)]] || <math>k((C_2)^4 : C_{15} \times C_2)</math> || 1 ||32 ||15 ||<math>C_{15}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,12)]] || <math>B_0(k(SL_2(16) \times C_2))</math> || 1 ||32 ||15 ||<math>C_{15}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,13)]] || <math>k(((C_2)^3:C_7) \times A_4)</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,14)]] || <math>B_0(k(((C_2)^3:C_7) \times A_5))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,15)]] || <math>B_0(k(SL_2(8) \times A_4))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,16)]] || <math>B_0(k(SL_2(8) \times A_5))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,17)]] || <math>k(((C_2)^3:(C_7:C_3)) \times (C_2)^2)</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,18)]] || <math>B_0(k(J_1 \times (C_2)^2))</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,19)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,20)]] || <math>k((C_2)^5:(C_7:C_3))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || The action of the subgroup <math>C_3</math> is as specified in M(32,51,4) above.<br />
|-<br />
|[[M(32,51,21)]] || <math>B_0(k((SL_2(8) \times (C_2)^2):C_3)</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || The action of the subgroup <math>C_3</math> is as specified in M(32,51,4) above.<br />
|-<br />
|[[M(32,51,22)]] || <math>k((C_2)^5:C_{31})</math> || 1 ||32 ||31 ||<math>C_{31}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,23)]] || <math>B_0(SL_2(32))</math> || 1 ||32 ||31 ||<math>C_{31}</math> ||<math>C_{5}</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,24)]] || <math>k(((C_2)^3:(C_7:C_3)) \times A_4)</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,25)]] || <math>B_0(k(((C_2)^3:(C_7:C_3)) \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,26)]] || <math>B_0(k(J_1 \times A_4))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,27)]] || <math>B_0(k(J_1 \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,28)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times A_4))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,29)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,30)]] || <math>k((C_2)^5:(C_{31}:C_5))</math> || 1 ||16 ||11 ||<math>C_{31}:C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,31)]] || <math>B_0({\rm Aut}(SL_2(32)))</math> || 1 ||16 ||11||<math>C_{31}:C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,32)]] || <math>b_2(k((C_2)^4 : 3^{1+2}_{+}) \times C_2)</math> || 1 ||16 ||1 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
|-<br />
|[[M(32,51,33)]] || <math>b_2(k((C_2)^5 : (C_7 : 3^{1+2}_{+})))</math> || 1 ||16 ||7 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
|-<br />
|[[M(32,51,34)]] || <math>b_2(k((SL_2(8) \times (C_2)^2) : 3^{1+2}_{+}))</math> || 1 ||16 ||7 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
<br />
|}<br />
<br />
If a block of a finite group is Morita equivalent to another block with defect group <math>(C_2)^5</math>, then it also has defect group <math>(C_2)^5</math>.<br />
<br />
Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [[References#A|[Ar19]]] and [[References#A|[AS20]]].<br />
<br />
All blocks with defect group <math>(C_2)^5</math> are derived equivalent to their Brauer correspondent, and each derived equivalence class is determined by the number of simple modules and the inertial quotient (and its action on the defect group) [[References#A|[AS20]]].<br />
<br />
It is unknown whether these derived equivalences are splendid, because we cannot say anything about the sources of the Morita equivalences determined in [[References#A|[Ar19]]].</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=(C2)%5E5&diff=1154(C2)^52020-11-13T09:40:08Z<p>CesareGArdito: </p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>(C_2)^5</math> ==<br />
<br />
These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]]<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]<br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(32,51,1)]] || <math>k((C_2)^5)</math> || 1 ||32 ||1 ||<math>1</math> || <math>(C_2)^5:GL_5(2)</math> || ||1 ||1 ||<br />
|-<br />
|[[M(32,51,2)]] || <math>k(A_4 \times (C_2)^3)</math> || 1 ||32 ||3 ||<math>C_3</math> || <math>((C_2)^3 : GL_3(2)) \times S_3</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,3)]] || <math>B_0(k(A_5 \times (C_2)^3))</math> || 1 ||32 ||3 ||<math>C_3</math> || <math>((C_2)^3 : GL_3(2)) \times C_2</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,4)]] || <math>k(((C_2)^4 :C_3) \times C_2)</math> || 1 ||16 ||3 ||<math>C_3</math> || || ||1 ||1 || The action of <math>C_3</math> comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math><br />
|-<br />
|[[M(32,51,5)]] || <math>k(((C_2)^4 :C_5) \times C_2)</math> || 1 ||16 ||5 ||<math>C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,6)]] || <math>k(((C_2)^3:C_7) \times (C_2)^2)</math> || 1 ||32 ||7 ||<math>C_7</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,7)]] || <math>B_0(k(SL_2(8) \times (C_2)^2))</math> || 1 ||32 ||7 ||<math>C_7</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,8)]] || <math>k(A_4 \times A_4 \times C_2)</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,9)]] || <math>B_0(k(A_4 \times A_5 \times C_2))</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||<br />
|-<br />
|[[M(32,51,10)]] || <math>B_0(k(A_5 \times A_5 \times C_2))</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,11)]] || <math>k((C_2)^4 : C_{15} \times C_2)</math> || 1 ||32 ||15 ||<math>C_{15}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,12)]] || <math>B_0(k(SL_2(16) \times C_2))</math> || 1 ||32 ||15 ||<math>C_{15}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,13)]] || <math>k(((C_2)^3:C_7) \times A_4)</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,14)]] || <math>B_0(k(((C_2)^3:C_7) \times A_5))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,15)]] || <math>B_0(k(SL_2(8) \times A_4))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,16)]] || <math>B_0(k(SL_2(8) \times A_5))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,17)]] || <math>k(((C_2)^3:(C_7:C_3)) \times (C_2)^2)</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,18)]] || <math>B_0(k(J_1 \times (C_2)^2))</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,19)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,20)]] || <math>k((C_2)^5:(C_7:C_3))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || The action of the subgroup <math>C_3</math> is as specified in M(32,51,4) above.<br />
|-<br />
|[[M(32,51,21)]] || <math>B_0(k((SL_2(8) \times (C_2)^2):C_3)</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || The action of the subgroup <math>C_3</math> is as specified in M(32,51,4) above.<br />
|-<br />
|[[M(32,51,22)]] || <math>k((C_2)^5:C_{31})</math> || 1 ||32 ||31 ||<math>C_{31}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,23)]] || <math>B_0(SL_2(32))</math> || 1 ||32 ||31 ||<math>C_{31}</math> ||<math>C_{5}</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,24)]] || <math>k(((C_2)^3:(C_7:C_3)) \times A_4)</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,25)]] || <math>B_0(k(((C_2)^3:(C_7:C_3)) \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,26)]] || <math>B_0(k(J_1 \times A_4))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,27)]] || <math>B_0(k(J_1 \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,28)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times A_4))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,29)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,30)]] || <math>k((C_2)^5:(C_{31}:C_5))</math> || 1 ||16 ||11 ||<math>C_{31}:C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,31)]] || <math>B_0({\rm Aut}(SL_2(32)))</math> || 1 ||16 ||11||<math>C_{31}:C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,32)]] || <math>b_2(k((C_2)^4 : 3^{1+2}_{+}) \times C_2)</math> || 1 ||16 ||1 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
|-<br />
|[[M(32,51,33)]] || <math>b_2(k((C_2)^5 : (C_7 : 3^{1+2}_{+})))</math> || 1 ||16 ||7 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
|-<br />
|[[M(32,51,34)]] || <math>b_2(k((SL_2(8) \times (C_2)^2) : 3^{1+2}_{+}))</math> || 1 ||16 ||7 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
<br />
|}<br />
<br />
If a block of a finite group is Morita equivalent to another block with defect group <math>(C_2)^5</math>, then it also has defect group <math>(C_2)^5</math>.<br />
<br />
Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [[References#A|[Ar19]]] and [[References#A|[AS20]]].<br />
<br />
All blocks with defect group <math>(C_2)^5</math> are derived equivalent to their Brauer correspondent, and each derived equivalence class is determined by the number of simple modules and the inertial quotient (and its action on the defect group) [[References#A|[AS20]]].<br />
<br />
It is unknown whether these derived equivalences are splendid, because we cannot say anything about the sources of the Morita equivalences determined in [[References#A|[Ar19]]].</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=(C2)%5E5&diff=1153(C2)^52020-11-13T09:39:25Z<p>CesareGArdito: /* Blocks with defect group (C_2)^5 */</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>(C_2)^5</math> ==<br />
<br />
These were classified in [[References#A|[Ar19]]] using the [[Glossary#CFSG|CFSG]]. Each of the 34 <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The known Picard groups were computed in [[References#E|[EL18c]]] or using the main theorem of [[References#L|[Liv19]]]<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]<br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(32,51,1)]] || <math>k((C_2)^5)</math> || 1 ||32 ||1 ||<math>1</math> || <math>(C_2)^5:GL_5(2)</math> || ||1 ||1 ||<br />
|-<br />
|[[M(32,51,2)]] || <math>k(A_4 \times (C_2)^3)</math> || 1 ||32 ||3 ||<math>C_3</math> || <math>((C_2)^3 : GL_3(2)) \times S_3</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,3)]] || <math>B_0(k(A_5 \times (C_2)^3))</math> || 1 ||32 ||3 ||<math>C_3</math> || <math>((C_2)^3 : GL_3(2)) \times C_2</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,4)]] || <math>k(((C_2)^4 :C_3) \times C_2)</math> || 1 ||16 ||3 ||<math>C_3</math> || || ||1 ||1 || The action of <math>C_3</math> comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math><br />
|-<br />
|[[M(32,51,5)]] || <math>k(((C_2)^4 :C_5) \times C_2)</math> || 1 ||16 ||5 ||<math>C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,6)]] || <math>k(((C_2)^3:C_7) \times (C_2)^2)</math> || 1 ||32 ||7 ||<math>C_7</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,7)]] || <math>B_0(k(SL_2(8) \times (C_2)^2))</math> || 1 ||32 ||7 ||<math>C_7</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,8)]] || <math>k(A_4 \times A_4 \times C_2)</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,9)]] || <math>B_0(k(A_4 \times A_5 \times C_2))</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||<br />
|-<br />
|[[M(32,51,10)]] || <math>B_0(k(A_5 \times A_5 \times C_2))</math> || 1 ||32 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,11)]] || <math>k((C_2)^4 : C_{15} \times C_2)</math> || 1 ||32 ||15 ||<math>C_{15}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,12)]] || <math>B_0(k(SL_2(16) \times C_2))</math> || 1 ||32 ||15 ||<math>C_{15}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,13)]] || <math>k(((C_2)^3:C_7) \times A_4)</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,14)]] || <math>B_0(k(((C_2)^3:C_7) \times A_5))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,15)]] || <math>B_0(k(SL_2(8) \times A_4))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,16)]] || <math>B_0(k(SL_2(8) \times A_5))</math> || 1 ||32 ||21 ||<math>C_{21}</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,17)]] || <math>k(((C_2)^3:(C_7:C_3)) \times (C_2)^2)</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,18)]] || <math>B_0(k(J_1 \times (C_2)^2))</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,19)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))</math> || 1 ||32 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(32,51,20)]] || <math>k((C_2)^5:(C_7:C_3))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || The action of the subgroup <math>C_3</math> is as specified in M(32,51,4) above.<br />
|-<br />
|[[M(32,51,21)]] || <math>B_0(k((SL_2(8) \times (C_2)^2):C_3)</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || The action of the subgroup <math>C_3</math> is as specified in M(32,51,4) above.<br />
|-<br />
|[[M(32,51,22)]] || <math>k((C_2)^5:C_{31})</math> || 1 ||32 ||31 ||<math>C_{31}</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,23)]] || <math>B_0(SL_2(32))</math> || 1 ||32 ||31 ||<math>C_{31}</math> ||<math>C_{5}</math> || ||1 ||1 || <br />
|-<br />
|[[M(32,51,24)]] || <math>k(((C_2)^3:(C_7:C_3)) \times A_4)</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,25)]] || <math>B_0(k(((C_2)^3:(C_7:C_3)) \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,26)]] || <math>B_0(k(J_1 \times A_4))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,27)]] || <math>B_0(k(J_1 \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,28)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times A_4))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,29)]] || <math>B_0(k({\rm Aut}(SL_2(8)) \times A_5))</math> || 1 ||32 ||15 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,30)]] || <math>k((C_2)^5:(C_{31}:C_5))</math> || 1 ||16 ||11 ||<math>C_{31}:C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,31)]] || <math>B_0({\rm Aut}(SL_2(32)))</math> || 1 ||16 ||11||<math>C_{31}:C_5</math> || || ||1 ||1 || <br />
|-<br />
|[[M(32,51,32)]] || <math>b_2(k((C_2)^4 : 3^{1+2}_{+}) \times C_2)</math> || 1 ||16 ||1 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
|-<br />
|[[M(32,51,33)]] || <math>b_2(k((C_2)^5 : (C_7 : 3^{1+2}_{+})))</math> || 1 ||16 ||7 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
|-<br />
|[[M(32,51,34)]] || <math>b_2(k((SL_2(8) \times (C_2)^2) : 3^{1+2}_{+}))</math> || 1 ||16 ||7 ||<math>(C_7:C_3) \times C_3</math> || || ||1 ||1 ||Non-principal block. Cannot be Morita equivalent to a principal block of any finite group.<br />
<br />
|}<br />
<br />
If a block of a finite group is Morita equivalent to another block with defect group <math>(C_2)^5</math>, then it also has defect group <math>(C_2)^5</math>.<br />
<br />
Blocks in the same Morita equivalence class have the same inertial quotient (with the same action on the defect group), as shown in [[References#A|[Ar19]]] and [[References#A|[AS20]]]<br />
<br />
All blocks with defect group <math>(C_2)^5</math> are derived equivalent to their Brauer correspondent, and each derived equivalence class is determined by the number of simple modules and the inertial quotient (and its action on the defect group) [[References#A|[AS20]]].<br />
<br />
It is unknown whether these derived equivalences are splendid, because we cannot say anything about the sources of the Morita equivalences determined in [[References#A|[Ar19]]].</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1152References2020-11-13T09:35:28Z<p>CesareGArdito: </p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
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|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Q8xC2&diff=1145Q8xC22020-08-15T22:31:04Z<p>CesareGArdito: /* Blocks with defect group Q_8 \times C_2 */</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>Q_8 \times C_2</math> ==<br />
<br />
The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13]]]. There is as yet no classification of blocks with these defect groups, but Donovan's conjecture is known to hold over <math>\mathcal{O}</math> [[References|[EL20]]].<br />
<br />
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''<br />
<br />
<!--{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| # lifts / <math>\mathcal{O}</math><br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||<br />
|-<br />
|[[M(16,9,2)]] || <math>B_0(k \tilde{S}_5)</math><ref>This is the double cover SmallGroup(240,89)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal A})</math><br />
|-<br />
|[[M(16,9,3)]] || <math>B_0(k \tilde{S}_4)</math><ref>This is the double cover SmallGroup(48,28)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal B})_1</math><br />
|-<br />
|[[M(16,9,4)]] || <math>B_0(kSL_2(9))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math><br />
|-<br />
|[[M(16,9,5)]] || <math>B_0(k(2.A_7))</math> || 1 ||10 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal B})</math><br />
|-<br />
|[[M(16,9,6)]] || <math>B_0(kSL_2(7))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal K})</math><br />
|}<br />
<br />
[[M(16,9,2)]] and [[M(16,9,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97]]], in which it is further proved that ''all'' blocks with defect group <math>Q_{16}</math> and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).<br />
<br />
[[M(16,9,4)]], [[M(16,9,5)]] and [[M(16,9,6)]] are derived equivalent over <math>\mathcal{O}</math> by [[References|[Ei16]]]<ref>This result was obtained over <math>k</math> in [[References|[Ho97]]]</ref>.--><br />
<br />
== Notes ==<br />
<br />
<references /></div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Q8xC2&diff=1144Q8xC22020-08-15T22:30:45Z<p>CesareGArdito: </p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>Q_8 \times C_2</math> ==<br />
<br />
The invariants <math>k(B)</math>, <math>k_i(B)</math> and <math>l(B)</math> are determined in [[References|[Sa13]]]. There is as yet no classification of blocks with these defect groups, but Donovan's conjecture is known to hold over <math>\mathcal{O}</math> [[References|[EaLi20]]].<br />
<br />
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''<br />
<br />
<!--{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| # lifts / <math>\mathcal{O}</math><br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||<br />
|-<br />
|[[M(16,9,2)]] || <math>B_0(k \tilde{S}_5)</math><ref>This is the double cover SmallGroup(240,89)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal A})</math><br />
|-<br />
|[[M(16,9,3)]] || <math>B_0(k \tilde{S}_4)</math><ref>This is the double cover SmallGroup(48,28)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal B})_1</math><br />
|-<br />
|[[M(16,9,4)]] || <math>B_0(kSL_2(9))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math><br />
|-<br />
|[[M(16,9,5)]] || <math>B_0(k(2.A_7))</math> || 1 ||10 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal B})</math><br />
|-<br />
|[[M(16,9,6)]] || <math>B_0(kSL_2(7))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal K})</math><br />
|}<br />
<br />
[[M(16,9,2)]] and [[M(16,9,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97]]], in which it is further proved that ''all'' blocks with defect group <math>Q_{16}</math> and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).<br />
<br />
[[M(16,9,4)]], [[M(16,9,5)]] and [[M(16,9,6)]] are derived equivalent over <math>\mathcal{O}</math> by [[References|[Ei16]]]<ref>This result was obtained over <math>k</math> in [[References|[Ho97]]]</ref>.--><br />
<br />
== Notes ==<br />
<br />
<references /></div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1112Generic classifications by p-group class2019-12-10T15:14:20Z<p>CesareGArdito: uniformed the style of 2 in the titles</p>
<hr />
<div>This page will contain results for generic classes of ''p''-groups. It is very much under construction so the list below is not complete.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most three ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
These have been classified in [[References#W|[WZZ18]]] and [[References#E|[EL18a]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math><br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 3</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|}<br />
<br />
== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.<br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References|[Mc19] ]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial, or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1111Generic classifications by p-group class2019-12-10T15:14:03Z<p>CesareGArdito: /* Abelian 2-groups */</p>
<hr />
<div>This page will contain results for generic classes of ''p''-groups. It is very much under construction so the list below is not complete.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most three ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
These have been classified in [[References#W|[WZZ18]]] and [[References#E|[EL18a]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math><br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 3</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|}<br />
<br />
== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.<br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic ''2''-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References|[Mc19] ]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial, or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1110Generic classifications by p-group class2019-12-10T15:13:52Z<p>CesareGArdito: /* Abelian 2-groups with 2-rank at most three */</p>
<hr />
<div>This page will contain results for generic classes of ''p''-groups. It is very much under construction so the list below is not complete.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most three ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
These have been classified in [[References#W|[WZZ18]]] and [[References#E|[EL18a]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math><br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 3</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|}<br />
<br />
== Abelian ''2''-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.<br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic ''2''-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References|[Mc19] ]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial, or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=To_do_list&diff=1107To do list2019-12-09T17:46:46Z<p>CesareGArdito: Updated list</p>
<hr />
<div>There's still a lot to do bring the site up to anywhere near the current state of knowledge. Here are some specific things which need doing. Also, many of the pages have been created quite quickly, and so accuracy checking is an important (and ongoing) task.<br />
<br />
*[[Blocks with cyclic defect groups]]. This background section has only just been started. Also [[Brauer trees]] page.<br />
*[[Tame blocks]]. This background section contains only basic information, and is not referenced properly yet.<br />
*Generic descriptions of tame blocks.<br />
*[[Basic algebras]] page creation. Will contain definition and overview.<br />
*Creation of class pages for known blocks with defect group [[C3xC3]].<br />
*Guralnick Ext space conjecture and status, including generalised (any pair of simple modules, bounded in terms of defect groups, as given in [[Statements of conjectures]]).<br />
*Generic classification for abelian 2-groups of rank at most 3, on [[Generic classifications by p-group class]] page. Including creation of group and class pages.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Notation&diff=1106Notation2019-12-09T17:45:39Z<p>CesareGArdito: fixed a typo</p>
<hr />
<div><math>(K,\mathcal{O},k)</math> is a <math>p</math>-modular system, where <math>\mathcal{O}</math> is a complete discrete valuation ring with algebraically closed residue field <math>k=\mathcal{O}/J(\mathcal{O})</math> and <math>K</math> is the field of fractions of <math>\mathcal{O}</math>, of characteristic zero. When we need to make a consistent choice of <math>(K,\mathcal{O},k)</math> we take <math>k</math> to be the algebraic closure of the field with <math>p</math> elements and <math>\mathcal{O}</math> to be the ring of Witt vectors for <math>k</math>. This has the disadvantage that for <math>G</math> a finite group <math>KG</math> need not contain the primitive character idempotents, but this condition can usually be avoided. In general however the choice of <math>\mathcal{O}</math> is not consistent across the literature and some care has to be taken.<br />
<br />
In the below, <math>G</math> is a finite group and <math>B</math> is a block of <math>\mathcal{O}G</math>. If it is clear from context, <math>B</math> may also mean the corresponding block of <math>kG</math>. When it is not otherwise clear from context <math>kB</math> will refer to the block of <math>kG</math>.<br />
<br />
{| <br />
|-<br />
|<math>k(B)</math> || Number of irreducible characters in <math>B</math>, equal to <math>\dim_k(Z(kB))</math> ||<br />
|-<br />
|<math>k_i(B)</math> || Number of irreducible characters in <math>B</math> of height <math>i</math> ||<br />
|-<br />
|<math>l(B)</math> || Number of isomorphism classes of simple <math>B</math>-modules ||<br />
|-<br />
| <math>{\rm mf_k(B)}</math> || The Morita-Frobenius number of <math>kB</math> || [[References|[Ke04] ]]<br />
|-<br />
|<math>{\rm mf_\mathcal{O}(B)}</math> || The <math>\mathcal{O}</math>-Morita Frobenius number ||<br />
|-<br />
|<math>{\rm Pic}_\mathcal{O}(B)</math> || The Picard group of <math>B</math> ||<br />
|- <br />
|<math>{\rm Pic}_k(B)</math> || The Picard group of <math>kB</math> ||<br />
|-<br />
|<math>\mathcal{T}(B)</math> || The subgroup of <math>{\rm Pic}_\mathcal{O}(B)</math> consisting of trivial source bimodules || [[References|[BKL18] ]]<br />
|-<br />
|<math>\mathcal{L}(B)</math> || The subgroup of <math>{\rm Pic}_\mathcal{O}(B)</math> consisting of linear source bimodules || [[References|[BKL18] ]]<br />
|-<br />
|<math>\mathcal{E}(B)</math> || The subgroup of <math>{\rm Pic}_\mathcal{O}(B)</math> consisting of endopermutation source bimodules || [[References|[BKL18] ]]<br />
|-<br />
|<math>M(x,y,z)</math> || A <math>k</math>-Morita equivalence class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class. <br />
|}</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(16,14,11)&diff=1105M(16,14,11)2019-12-09T17:40:46Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(16,14,11) - <math>k((C_2)^4 : C_{15})</math> <br />
|image = &nbsp; <br />
|representative = <math>k((C_2)^4 : C_{15})</math><br />
|defect = [[(C2)%5E4|<math>(C_2)^4</math>]]<br />
|inertialquotients = <math>C_{15}</math><br />
|k(B) = 16<br />
|l(B) = 15<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O} ((C_2)^4 : C_5)</math><br />
|decomp = See below<br />
|O-morita-frob = 1<br />
|Pic-O = <math> C_{15} : C_4</math><br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(16,14,12)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
The action of <math>C_{15}</math> on the defect group is the one of a Singer cycle.<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(16,14,11), then <math>B</math> is in [[M(16,14,4)]], [[M(16,14,5)]] or M(16,14,11).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccccccccccccccc}<br />
2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2<br />
\end{array}\right)</math><br />
<br />
<br />
== Decomposition matrix ==<br />
<math>\left( \begin{array}{ccccccccccccccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1<br />
\end{array}\right)</math><br />
<br />
<br />
<br />
[[(C2)%5E4|Back to <math>(C_2)^4</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(16,14,12)&diff=1104M(16,14,12)2019-12-09T17:40:35Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(16,14,12) - <math>B_0(k(SL_2(16)))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k(SL_2(16)))</math><br />
|defect = [[(C2)%5E4|<math>(C_2)^4</math>]]<br />
|inertialquotients = <math>C_{15}</math><br />
|k(B) = 16<br />
|l(B) = 15<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}(SL_2(16)))</math><br />
|decomp = See below<br />
|O-morita-frob = 1<br />
|Pic-O = <math>C_4</math><br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(16,14,11)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(16,14,12), then <math>B</math> is also in M(16,14,12).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccccccc}<br />
16 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 2 \\<br />
8 & 8 & 4 & 4 & 4 & 2 & 4 & 2 & 2 & 4 & 0 & 2 & 0 & 0 & 1 \\<br />
8 & 4 & 8 & 4 & 4 & 4 & 2 & 0 & 2 & 2 & 4 & 1 & 2 & 0 & 0 \\<br />
8 & 4 & 4 & 8 & 4 & 2 & 4 & 4 & 0 & 2 & 2 & 0 & 1 & 2 & 0 \\<br />
8 & 4 & 4 & 4 & 8 & 4 & 2 & 2 & 4 & 0 & 2 & 0 & 0 & 1 & 2 \\<br />
4 & 2 & 4 & 2 & 4 & 4 & 1 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 \\<br />
4 & 4 & 2 & 4 & 2 & 1 & 4 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\<br />
4 & 2 & 0 & 4 & 2 & 0 & 2 & 4 & 0 & 1 & 0 & 0 & 0 & 2 & 0 \\<br />
4 & 2 & 2 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 1 & 0 & 0 & 0 & 2 \\<br />
4 & 4 & 2 & 2 & 0 & 0 & 2 & 1 & 0 & 4 & 0 & 2 & 0 & 0 & 0 \\<br />
4 & 0 & 4 & 2 & 2 & 2 & 0 & 0 & 1 & 0 & 4 & 0 & 2 & 0 & 0 \\<br />
2 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 \\<br />
2 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 \\<br />
2 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\<br />
2 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2<br />
\end{array}\right)</math><br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccccccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E4|Back to <math>(C_2)^4</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=(C2)%5E4&diff=1103(C2)^42019-12-09T17:39:54Z<p>CesareGArdito: Removed possibly incorrect Picard groups</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>(C_2)^4</math> ==<br />
<br />
These were classified in [[References#E|[Ea18]]] using the [[Glossary#CFSG|CFSG]]. Each of the sixteen <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The possibilities for <math> k(B) \text{ and } l(B)</math> were computed in [[References#K|[KS13]]] and [[References#E|[EKKS14]]].<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]<br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,14,1)]] || <math>k((C_2)^4)</math> || 1 ||16 ||1 ||<math>1</math> || <math>(C_2)^4:GL_4(2)</math> || ||1 ||1 ||<br />
|-<br />
|[[M(16,14,2)]] || <math>B_0(k(C_2 \times C_2 \times A_5))</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times C_2</math> || ||1 ||1 || <br />
|-<br />
|[[M(16,14,3)]] || <math>k(C_2 \times C_2 \times A_4)</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times S_3</math> || ||1 ||1 || <br />
|-<br />
|[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math><br />
|-<br />
|[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || || ||1 ||1 || The action comes from the 3rd power of a Singer cycle for <math>\mathbb{F}_{16}</math><br />
|-<br />
|[[M(16,14,6)]] || <math>k(C_2 \times ((C_2)^3:C_7))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,14,7)]] || <math>B_0(k(C_2 \times SL_2(8)))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(16,14,8)]] || <math>k(A_4 \times A_4)</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || <math>S_3 \wr C_2</math>|| ||1 ||1 || <br />
|-<br />
|[[M(16,14,9)]] || <math>B_0(k(A_4 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || <math>S_3 \times C_2</math> || ||1 ||1 ||<br />
|-<br />
|[[M(16,14,10)]] || <math>B_0(k(A_5 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}:C_4</math> || ||1 ||1 || <br />
|-<br />
|[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_4</math> || ||1 ||1 || <br />
|-<br />
|[[M(16,14,13)]] || <math>k(C_2 \times ((C_2)^3:(C_7:C_3)))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,14,14)]] || <math>B_0(k(C_2 \times J_1))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,14,15)]] || <math>B_0(k(C_2 \times{\rm Aut}(SL_2(8))))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(16,14,16)]] || <math>b_2(k((C_2)^4 : 3^{1+2}_{+}))</math> || 1 ||8 ||1 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||Non-principal faithful block. Cannot be Morita equivalent to a principal block of any finite group.<br />
|}<br />
<br />
Both non-principal faithful blocks of <math>k((C_2)^4 : 3^{1+2}_{+})</math> and <math>k((C_2)^4 : 3^{1+2}_{-})</math> are Morita equivalent.<br />
<br />
Blocks are derived equivalent if and only if they have the same inertial quotient (with the same action on the defect group) and number of simple modules. All the derived equivalences here also occur over <math>\mathcal{O}</math>.<br />
In particular: <br />
<br />
[[M(16,14,2)]] and [[M(16,14,3)]] are derived equivalent over <math>\mathcal{O}</math>.<br />
<br />
[[M(16,14,6)]] and [[M(16,14,7)]] are derived equivalent over <math>\mathcal{O}</math>.<br />
<br />
[[M(16,14,8)]] [[M(16,14,9)]] and [[M(16,14,10)]] are derived equivalent over <math>\mathcal{O}</math>.<br />
<br />
[[M(16,14,11)]] and [[M(16,14,12)]] are derived equivalent over <math>\mathcal{O}</math>.<br />
<br />
[[M(16,14,13)]], [[M(16,14,14)]] and [[M(16,14,15)]] are derived equivalent over <math>\mathcal{O}</math>.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=(C2)%5E4&diff=1102(C2)^42019-12-09T17:38:12Z<p>CesareGArdito: /* Blocks with defect group (C_2)^4 */</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>(C_2)^4</math> ==<br />
<br />
These were classified in [[References#E|[Ea18]]] using the [[Glossary#CFSG|CFSG]]. Each of the sixteen <math>k</math>-Morita equivalence classes lifts to an unique class over <math>\mathcal{O}</math>. The possibilities for <math> k(B) \text{ and } l(B)</math> were computed in [[References#K|[KS13]]] and [[References#E|[EKKS14]]].<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]<br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,14,1)]] || <math>k((C_2)^4)</math> || 1 ||16 ||1 ||<math>1</math> || <math>(C_2)^4:GL_4(2)</math> || ||1 ||1 ||<br />
|-<br />
|[[M(16,14,2)]] || <math>B_0(k(C_2 \times C_2 \times A_5))</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times C_2</math> || ||1 ||1 || <br />
|-<br />
|[[M(16,14,3)]] || <math>k(C_2 \times C_2 \times A_4)</math> || 1 ||16 ||3 ||<math>C_3</math> || <math>((C_2 \times C_2):S_3) \times S_3</math> || ||1 ||1 || <br />
|-<br />
|[[M(16,14,4)]] || <math>k((C_2)^4 :C_3)</math> || 1 ||8 ||3 ||<math>C_3</math> || <math>C_3</math> || ||1 ||1 || The action comes from the 5th power of a Singer cycle for <math>\mathbb{F}_{16}</math><br />
|-<br />
|[[M(16,14,5)]] || <math>k((C_2)^4 : C_5)</math> || 1 ||8 ||5 ||<math>C_5</math> || <math>C_5</math> || ||1 ||1 || The action comes from the 3rd power of a Singer cycle for <math>\mathbb{F}_{16}</math><br />
|-<br />
|[[M(16,14,6)]] || <math>k(C_2 \times ((C_2)^3:C_7))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,14,7)]] || <math>B_0(k(C_2 \times SL_2(8)))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(16,14,8)]] || <math>k(A_4 \times A_4)</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || <math>S_3 \wr C_2</math>|| ||1 ||1 || <br />
|-<br />
|[[M(16,14,9)]] || <math>B_0(k(A_4 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || <math>S_3 \times C_2</math> || ||1 ||1 ||<br />
|-<br />
|[[M(16,14,10)]] || <math>B_0(k(A_5 \times A_5))</math> || 1 ||16 ||9 ||<math>C_3 \times C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}:C_4</math> || ||1 ||1 || <br />
|-<br />
|[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_4</math> || ||1 ||1 || <br />
|-<br />
|[[M(16,14,13)]] || <math>k(C_2 \times ((C_2)^3:(C_7:C_3)))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,14,14)]] || <math>B_0(k(C_2 \times J_1))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,14,15)]] || <math>B_0(k(C_2 \times{\rm Aut}(SL_2(8))))</math> || 1 ||16 ||5 ||<math>C_7:C_3</math> || || ||1 ||1 ||<br />
|- <br />
|[[M(16,14,16)]] || <math>b_2(k((C_2)^4 : 3^{1+2}_{+}))</math> || 1 ||8 ||1 ||<math>C_3 \times C_3</math> || || ||1 ||1 ||Non-principal faithful block. Cannot be Morita equivalent to a principal block of any finite group.<br />
|}<br />
<br />
Both non-principal faithful blocks of <math>k((C_2)^4 : 3^{1+2}_{+})</math> and <math>k((C_2)^4 : 3^{1+2}_{-})</math> are Morita equivalent.<br />
<br />
Blocks are derived equivalent if and only if they have the same inertial quotient (with the same action on the defect group) and number of simple modules. All the derived equivalences here also occur over <math>\mathcal{O}</math>.<br />
In particular: <br />
<br />
[[M(16,14,2)]] and [[M(16,14,3)]] are derived equivalent over <math>\mathcal{O}</math>.<br />
<br />
[[M(16,14,6)]] and [[M(16,14,7)]] are derived equivalent over <math>\mathcal{O}</math>.<br />
<br />
[[M(16,14,8)]] [[M(16,14,9)]] and [[M(16,14,10)]] are derived equivalent over <math>\mathcal{O}</math>.<br />
<br />
[[M(16,14,11)]] and [[M(16,14,12)]] are derived equivalent over <math>\mathcal{O}</math>.<br />
<br />
[[M(16,14,13)]], [[M(16,14,14)]] and [[M(16,14,15)]] are derived equivalent over <math>\mathcal{O}</math>.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1101Guide to contributing: Code2019-12-09T17:16:31Z<p>CesareGArdito: </p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="overflow:auto;"><br />
<div style="font-weight:bold;line-height:1.6;">Magma code</div><br />
<div class="mw-collapsible-content"><br />
<pre><br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End of input data*/<br />
<br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
</div></div><br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.<br />
<br />
== GAP Code ==</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1100Guide to contributing: Code2019-12-09T17:16:10Z<p>CesareGArdito: </p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="overflow:auto;"><br />
<div style="font-weight:bold;line-height:1.6;">Magma code</div><br />
<div class="mw-collapsible-content"><br />
<pre><br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End of input data*/<br />
<br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
</div></div><br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1099Guide to contributing: Code2019-12-09T17:15:49Z<p>CesareGArdito: </p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="overflow:auto;"><br />
<div style="font-weight:bold;line-height:1.6;">Magma code</div><br />
<pre><br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End of input data*/<br />
<br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
</div></div><br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1098Guide to contributing: Code2019-12-09T17:15:33Z<p>CesareGArdito: /* Magma code */</p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="overflow:auto;"><br />
<div style="font-weight:bold;line-height:1.6;">Magma code</div><br />
<pre><br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End of input data*/<br />
<br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
</div><br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1097Guide to contributing: Code2019-12-09T17:15:07Z<p>CesareGArdito: </p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="overflow:auto;"><br />
<pre><br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End of input data*/<br />
<br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
</div><br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1096Guide to contributing: Code2019-12-09T17:14:12Z<p>CesareGArdito: </p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<div class="toccolours mw-collapsible" style="overflow:auto;"><br />
<pre><br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End of input data*/<br />
<br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
</div><br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1095Guide to contributing: Code2019-12-09T16:50:52Z<p>CesareGArdito: /* Magma code */</p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<pre><br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End of input data*/<br />
<br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
<br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1094Guide to contributing: Code2019-12-09T16:48:44Z<p>CesareGArdito: </p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<pre><br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End input data*/<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
<br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1093Guide to contributing: Code2019-12-09T16:48:37Z<p>CesareGArdito: </p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator[http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<pre><br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End input data*/<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
<br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1092Guide to contributing: Code2019-12-09T16:47:49Z<p>CesareGArdito: </p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free [[http://magma.maths.usyd.edu.au/calc/ Magma online calculator]], as long as the involved group is small.<br />
<br />
<pre><br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End input data*/<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
<br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1091Guide to contributing: Code2019-12-09T16:47:09Z<p>CesareGArdito: Created page with "The code in this section computes the Cartan. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page. == Magma code == The follow..."</p>
<hr />
<div>The code in this section computes the Cartan. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free [[http://magma.maths.usyd.edu.au/calc/ Magma online calculator]], as long as the involved group is small.<br />
<br />
<pre><br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End input data*/<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
<br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing&diff=1090Guide to contributing2019-12-09T16:32:41Z<p>CesareGArdito: </p>
<hr />
<div>At present you must be registered to edit this site. Eventually we hope to make editing open to all. Please contact [[User:Mcbssce|Charles Eaton]] to register.<br />
<br />
Consult the [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents User's Guide] for information on using the wiki software.<br />
<br />
There is a [[Sandbox]] to experiment with editing.<br />
<br />
When referencing a paper, please use the labelling in [[References]], and make the appropriate entry in [[References]]. When referencing, use, e.g., <pre>[[References#T|[TH85]]]</pre> which will automatically link to the start of the "T"s in the reference list.<br />
<br />
The following is a suggested template for creating a page for a Morita equivalence class. The blockbox part should be used as given. See [[Labelling for Morita equivalence classes|this page]] for the labelling system for Morita equivalence classes. Please try to follow existing classifications for labelling where possible, for example for tame blocks try to use the ordering used in Chapter 10 of [[References|[Er90] ]].<br />
<br />
<pre>{{blockbox<br />
|title = M(x,y,z) - <math>INSERT / EXAMPLE / HERE</math> <br />
|image = <br />
|representative = <br />
|defect =<br />
|inertialquotients =<br />
|k(B) =<br />
|l(B) =<br />
|k-morita-frob = <br />
|Pic-k=<br />
|cartan = <br />
|defect-morita-inv? =<br />
|inertial-morita-inv? =<br />
|O-morita? = <br />
|O-morita =<br />
|decomp =<br />
|O-morita-frob =<br />
|Pic-O =<br />
|PIgroup =<br />
|source? =<br />
|sourcereps =<br />
|k-derived-known? =<br />
|k-derived =<br />
|O-derived-known? = <br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks = <br />
}}<br />
<br />
SOME BRIEF INTRODUCTION HERE<br />
<br />
== Basic algebra ==<br />
<br />
'''Quiver:'''<br />
<br />
'''Relations w.r.t. <math>k</math>:'''<br />
<br />
== Other notatable representatives ==<br />
<br />
== Projective indecomposable modules ==<br />
<br />
== Irreducible characters ==<br />
<br />
INFORMATION ON HEIGHTS OF IRREDUCIBLE CHARACTERS, PLUS OTHER INTERESTING INFORMATION.<br />
<br />
== Notes ==<br />
<br />
<references /><br />
<br />
FOOTNOTES WILL APPEAR HERE AUTOMATICALLY IF YOU PUT <ref>xxxx</ref> IN THE MAIN TEXT.<br />
</pre><br />
<br />
<br />
== Code ==<br />
It is possible to use the code in [[Guide to contributing: Code]] in order to compute the Cartan matrix, decomposition matrices and the structure of the projective indecomposable modules of a given group algebra (and hence of its blocks).<br />
<br />
== Getting started ==<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]<br />
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]<br />
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,31)&diff=1089M(32,51,31)2019-12-09T16:15:20Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,31) - <math>B_0(k({\rm Aut}SL_2(32)))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k({\rm Aut}SL_2(32)))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{31}:C_5</math><br />
|k(B) = 16<br />
|l(B) = 11<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}({\rm Aut}SL_2(32)))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,30)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,31), then <math>B</math> is in [[M(32,51,23)]] or M(32,51,31).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{11}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{2} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{2} S_{2} S_{1} S_{4} S_{3} S_{5} S_{1} S_{3} S_{5} S_{4} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{5} S_{1} S_{3} S_{4} S_{3} S_{1} S_{2} S_{4} S_{2} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{6} \\<br />
S_{4} S_{3} S_{2} S_{5} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{3} S_{2} S_{2} S_{3} S_{4} S_{1} S_{1} S_{5} S_{7} S_{8} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{2} S_{5} S_{4} S_{4} S_{1} S_{1} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{5} S_{2} S_{3} S_{1} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{5} S_{2} S_{1} S_{4} S_{4} S_{2} S_{3} S_{3} S_{8} S_{8} S_{7} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{3} S_{3} S_{5} S_{5} S_{2} S_{1} S_{2} S_{4} S_{1} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{1} S_{3} S_{5} S_{2} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{3} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{6} \\<br />
S_{3} S_{2} S_{5} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{1} S_{5} S_{2} S_{2} S_{3} S_{3} S_{1} S_{5} S_{4} S_{4} S_{7} S_{8} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{3} S_{1} S_{4} S_{4} S_{2} S_{2} S_{1} S_{5} S_{3} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{6} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{4} S_{1} S_{5} S_{2} S_{1} S_{2} S_{3} S_{8} S_{7} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{4} S_{2} S_{2} S_{3} S_{5} S_{5} S_{1} S_{3} S_{1} S_{4} S_{7} S_{7} S_{8} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{2} S_{5} S_{3} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{5} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{4} S_{1} S_{3} S_{2} S_{5} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{1} S_{1} S_{3} S_{3} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{2} S_{5} S_{4} S_{1} S_{5} S_{2} S_{4} S_{4} S_{4} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{1} S_{2} S_{3} S_{4} S_{5} S_{5} S_{2} S_{5} S_{2} S_{5} S_{2} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{3} S_{4} S_{5} S_{4} S_{1} S_{2} S_{1} S_{4} S_{3} S_{1} S_{1} S_{7} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{7} S_{7} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{2} S_{4} S_{5} S_{4} S_{5} S_{4} S_{3} S_{4} S_{3} S_{2} S_{3} S_{1} S_{3} S_{5} S_{2} S_{1} S_{1} S_{5} S_{1} S_{2} S_{7} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{2} S_{1} S_{5} S_{4} S_{3} S_{8} S_{8} S_{7} \\<br />
S_{6} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{4} S_{3} S_{2} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{4} S_{5} S_{1} S_{5} S_{3} S_{4} S_{2} S_{5} S_{1} S_{2} S_{3} S_{3} S_{2} S_{1} S_{7} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} \\<br />
S_{1} S_{2} S_{1} S_{1} S_{4} S_{4} S_{2} S_{2} S_{5} S_{3} S_{4} S_{3} S_{3} S_{5} S_{5} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{1} S_{3} S_{5} S_{4} S_{2} S_{7} S_{7} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{1} S_{4} S_{5} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{2} S_{4} S_{5} S_{5} S_{4} S_{2} S_{1} S_{1} S_{3} S_{1} S_{3} S_{3} S_{2} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{10} \\<br />
S_{4} S_{4} S_{4} S_{2} S_{1} S_{1} S_{5} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{5} S_{3} S_{8} S_{8} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{4} S_{1} S_{2} S_{8} S_{7} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{8} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{7} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{3} S_{1} S_{4} S_{5} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{2} S_{5} S_{3} S_{5} S_{1} S_{4} S_{3} S_{4} S_{1} S_{2} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{4} S_{3} S_{2} S_{1} S_{5} S_{8} S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{7} S_{11} \\<br />
S_{9} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{2} S_{4} S_{3} S_{1} S_{5} S_{8} S_{7} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{1} S_{3} S_{3} S_{5} S_{5} S_{2} S_{4} S_{4} S_{1} S_{2} S_{8} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{3} S_{2} S_{1} S_{5} S_{4} S_{7} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{8} S_{7} \\<br />
S_{10} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{9} \\<br />
S_{7} \\<br />
S_{6} \\<br />
S_{1} S_{4} S_{3} S_{5} S_{2} \\<br />
S_{6} \\<br />
S_{2} S_{1} S_{3} S_{4} S_{5} \\<br />
S_{6} \\<br />
S_{7} \\<br />
S_{9} \\<br />
S_{11} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
8 & 6 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 8 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 8 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 8 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 6 & 8 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
16 & 16 & 16 & 16 & 16 & 48 & 20 & 28 & 8 & 12 & 3 \\<br />
8 & 8 & 8 & 8 & 8 & 20 & 12 & 10 & 5 & 6 & 2 \\<br />
8 & 8 & 8 & 8 & 8 & 28 & 10 & 20 & 2 & 7 & 0 \\<br />
4 & 4 & 4 & 4 & 4 & 8 & 5 & 2 & 4 & 2 & 2 \\<br />
4 & 4 & 4 & 4 & 4 & 12 & 6 & 7 & 2 & 4 & 0 \\<br />
2 & 2 & 2 & 2 & 2 & 3 & 2 & 0 & 2 & 0 & 2<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 4 & 1 & 3 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 1 & 0 & 1 & 0 & 1<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,31)&diff=1088M(32,51,31)2019-12-09T16:13:15Z<p>CesareGArdito: /* Projective indecomposable modules */</p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,32) - <math>B_0(k({\rm Aut}SL_2(32)))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k({\rm Aut}SL_2(32)))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{31}:C_5</math><br />
|k(B) = 16<br />
|l(B) = 11<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}({\rm Aut}SL_2(32)))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,30)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,31), then <math>B</math> is in [[M(32,51,23)]] or M(32,51,31).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{11}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{2} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{2} S_{2} S_{1} S_{4} S_{3} S_{5} S_{1} S_{3} S_{5} S_{4} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{5} S_{1} S_{3} S_{4} S_{3} S_{1} S_{2} S_{4} S_{2} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{6} \\<br />
S_{4} S_{3} S_{2} S_{5} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{3} S_{2} S_{2} S_{3} S_{4} S_{1} S_{1} S_{5} S_{7} S_{8} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{2} S_{5} S_{4} S_{4} S_{1} S_{1} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{5} S_{2} S_{3} S_{1} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{5} S_{2} S_{1} S_{4} S_{4} S_{2} S_{3} S_{3} S_{8} S_{8} S_{7} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{3} S_{3} S_{5} S_{5} S_{2} S_{1} S_{2} S_{4} S_{1} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{1} S_{3} S_{5} S_{2} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{3} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{6} \\<br />
S_{3} S_{2} S_{5} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{1} S_{5} S_{2} S_{2} S_{3} S_{3} S_{1} S_{5} S_{4} S_{4} S_{7} S_{8} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{3} S_{1} S_{4} S_{4} S_{2} S_{2} S_{1} S_{5} S_{3} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{6} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{4} S_{1} S_{5} S_{2} S_{1} S_{2} S_{3} S_{8} S_{7} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{4} S_{2} S_{2} S_{3} S_{5} S_{5} S_{1} S_{3} S_{1} S_{4} S_{7} S_{7} S_{8} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{2} S_{5} S_{3} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{5} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{4} S_{1} S_{3} S_{2} S_{5} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{1} S_{1} S_{3} S_{3} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{2} S_{5} S_{4} S_{1} S_{5} S_{2} S_{4} S_{4} S_{4} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{1} S_{2} S_{3} S_{4} S_{5} S_{5} S_{2} S_{5} S_{2} S_{5} S_{2} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{3} S_{4} S_{5} S_{4} S_{1} S_{2} S_{1} S_{4} S_{3} S_{1} S_{1} S_{7} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{7} S_{7} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{2} S_{4} S_{5} S_{4} S_{5} S_{4} S_{3} S_{4} S_{3} S_{2} S_{3} S_{1} S_{3} S_{5} S_{2} S_{1} S_{1} S_{5} S_{1} S_{2} S_{7} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{2} S_{1} S_{5} S_{4} S_{3} S_{8} S_{8} S_{7} \\<br />
S_{6} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{4} S_{3} S_{2} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{4} S_{5} S_{1} S_{5} S_{3} S_{4} S_{2} S_{5} S_{1} S_{2} S_{3} S_{3} S_{2} S_{1} S_{7} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} \\<br />
S_{1} S_{2} S_{1} S_{1} S_{4} S_{4} S_{2} S_{2} S_{5} S_{3} S_{4} S_{3} S_{3} S_{5} S_{5} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{1} S_{3} S_{5} S_{4} S_{2} S_{7} S_{7} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{1} S_{4} S_{5} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{2} S_{4} S_{5} S_{5} S_{4} S_{2} S_{1} S_{1} S_{3} S_{1} S_{3} S_{3} S_{2} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{10} \\<br />
S_{4} S_{4} S_{4} S_{2} S_{1} S_{1} S_{5} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{5} S_{3} S_{8} S_{8} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{4} S_{1} S_{2} S_{8} S_{7} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{8} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{7} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{3} S_{1} S_{4} S_{5} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{2} S_{5} S_{3} S_{5} S_{1} S_{4} S_{3} S_{4} S_{1} S_{2} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{4} S_{3} S_{2} S_{1} S_{5} S_{8} S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{7} S_{11} \\<br />
S_{9} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{2} S_{4} S_{3} S_{1} S_{5} S_{8} S_{7} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{1} S_{3} S_{3} S_{5} S_{5} S_{2} S_{4} S_{4} S_{1} S_{2} S_{8} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{3} S_{2} S_{1} S_{5} S_{4} S_{7} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{8} S_{7} \\<br />
S_{10} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{9} \\<br />
S_{7} \\<br />
S_{6} \\<br />
S_{1} S_{4} S_{3} S_{5} S_{2} \\<br />
S_{6} \\<br />
S_{2} S_{1} S_{3} S_{4} S_{5} \\<br />
S_{6} \\<br />
S_{7} \\<br />
S_{9} \\<br />
S_{11} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
8 & 6 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 8 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 8 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 8 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 6 & 8 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
16 & 16 & 16 & 16 & 16 & 48 & 20 & 28 & 8 & 12 & 3 \\<br />
8 & 8 & 8 & 8 & 8 & 20 & 12 & 10 & 5 & 6 & 2 \\<br />
8 & 8 & 8 & 8 & 8 & 28 & 10 & 20 & 2 & 7 & 0 \\<br />
4 & 4 & 4 & 4 & 4 & 8 & 5 & 2 & 4 & 2 & 2 \\<br />
4 & 4 & 4 & 4 & 4 & 12 & 6 & 7 & 2 & 4 & 0 \\<br />
2 & 2 & 2 & 2 & 2 & 3 & 2 & 0 & 2 & 0 & 2<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 4 & 1 & 3 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 1 & 0 & 1 & 0 & 1<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,31)&diff=1087M(32,51,31)2019-12-09T16:12:51Z<p>CesareGArdito: /* Projective indecomposable modules */</p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,32) - <math>B_0(k({\rm Aut}SL_2(32)))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k({\rm Aut}SL_2(32)))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{31}:C_5</math><br />
|k(B) = 16<br />
|l(B) = 11<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}({\rm Aut}SL_2(32)))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,30)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,31), then <math>B</math> is in [[M(32,51,23)]] or M(32,51,31).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{11}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{2} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{2} S_{2} S_{1} S_{4} S_{3} S_{5} S_{1} S_{3} S_{5} S_{4} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{5} S_{1} S_{3} S_{4} S_{3} S_{1} S_{2} S_{4} S_{2} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{6} \\<br />
S_{4} S_{3} S_{2} S_{5} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{3} S_{2} S_{2} S_{3} S_{4} S_{1} S_{1} S_{5} S_{7} S_{8} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{2} S_{5} S_{4} S_{4} S_{1} S_{1} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{5} S_{2} S_{3} S_{1} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{5} S_{2} S_{1} S_{4} S_{4} S_{2} S_{3} S_{3} S_{8} S_{8} S_{7} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{3} S_{3} S_{5} S_{5} S_{2} S_{1} S_{2} S_{4} S_{1} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{1} S_{3} S_{5} S_{2} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{3} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{6} \\<br />
S_{3} S_{2} S_{5} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{1} S_{5} S_{2} S_{2} S_{3} S_{3} S_{1} S_{5} S_{4} S_{4} S_{7} S_{8} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{3} S_{1} S_{4} S_{4} S_{2} S_{2} S_{1} S_{5} S_{3} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{6} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{4} S_{1} S_{5} S_{2} S_{1} S_{2} S_{3} S_{8} S_{7} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{4} S_{2} S_{2} S_{3} S_{5} S_{5} S_{1} S_{3} S_{1} S_{4} S_{7} S_{7} S_{8} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{2} S_{5} S_{3} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{5} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{4} S_{1} S_{3} S_{2} S_{5} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{1} S_{1} S_{3} S_{3} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{2} S_{5} S_{4} S_{1} S_{5} S_{2} S_{4} S_{4} S_{4} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{1} S_{2} S_{3} S_{4} S_{5} S_{5} S_{2} S_{5} S_{2} S_{5} S_{2} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{3} S_{4} S_{5} S_{4} S_{1} S_{2} S_{1} S_{4} S_{3} S_{1} S_{1} S_{7} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{7} S_{7} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{2} S_{4} S_{5} S_{4} S_{5} S_{4} S_{3} S_{4} S_{3} S_{2} S_{3} S_{1} S_{3} S_{5} S_{2} S_{1} S_{1} S_{5} S_{1} S_{2} S_{7} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{2} S_{1} S_{5} S_{4} S_{3} S_{8} S_{8} S_{7} \\<br />
S_{6} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{4} S_{3} S_{2} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{4} S_{5} S_{1} S_{5} S_{3} S_{4} S_{2} S_{5} S_{1} S_{2} S_{3} S_{3} S_{2} S_{1} S_{7} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} \\<br />
S_{1} S_{2} S_{1} S_{1} S_{4} S_{4} S_{2} S_{2} S_{5} S_{3} S_{4} S_{3} S_{3} S_{5} S_{5} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{1} S_{3} S_{5} S_{4} S_{2} S_{7} S_{7} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{1} S_{4} S_{5} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{2} S_{4} S_{5} S_{5} S_{4} S_{2} S_{1} S_{1} S_{3} S_{1} S_{3} S_{3} S_{2} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{10} \\<br />
S_{4} S_{4} S_{4} S_{2} S_{1} S_{1} S_{5} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{5} S_{3} S_{8} S_{8} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{4} S_{1} S_{2} S_{8} S_{7} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{8} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{7} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{3} S_{1} S_{4} S_{5} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{2} S_{5} S_{3} S_{5} S_{1} S_{4} S_{3} S_{4} S_{1} S_{2} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{4} S_{3} S_{2} S_{1} S_{5} S_{8} S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{7} S_{11} \\<br />
S_{9} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{2} S_{4} S_{3} S_{1} S_{5} S_{8} S_{7} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{1} S_{3} S_{3} S_{5} S_{5} S_{2} S_{4} S_{4} S_{1} S_{2} S_{8} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{3} S_{2} S_{1} S_{5} S_{4} S_{7} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{8} S_{7} \\<br />
S_{10} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{9} \\<br />
S_{7} \\<br />
S_{6} \\<br />
S_{1} S_{4} S_{3} S_{5} S_{2} \\<br />
S_{6} \\<br />
S_{2} S_{1} S_{3} S_{4} S_{5} \\<br />
S_{6} \\<br />
S_{7} \\<br />
S_{9} \\<br />
S_{11} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
8 & 6 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 8 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 8 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 8 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 6 & 8 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
16 & 16 & 16 & 16 & 16 & 48 & 20 & 28 & 8 & 12 & 3 \\<br />
8 & 8 & 8 & 8 & 8 & 20 & 12 & 10 & 5 & 6 & 2 \\<br />
8 & 8 & 8 & 8 & 8 & 28 & 10 & 20 & 2 & 7 & 0 \\<br />
4 & 4 & 4 & 4 & 4 & 8 & 5 & 2 & 4 & 2 & 2 \\<br />
4 & 4 & 4 & 4 & 4 & 12 & 6 & 7 & 2 & 4 & 0 \\<br />
2 & 2 & 2 & 2 & 2 & 3 & 2 & 0 & 2 & 0 & 2<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 4 & 1 & 3 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 1 & 0 & 1 & 0 & 1<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,31)&diff=1086M(32,51,31)2019-12-09T16:12:35Z<p>CesareGArdito: /* Projective indecomposable modules */</p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,32) - <math>B_0(k({\rm Aut}SL_2(32)))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k({\rm Aut}SL_2(32)))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{31}:C_5</math><br />
|k(B) = 16<br />
|l(B) = 11<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}({\rm Aut}SL_2(32)))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,30)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,31), then <math>B</math> is in [[M(32,51,23)]] or M(32,51,31).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{11}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{2} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{2} S_{2} S_{1} S_{4} S_{3} S_{5} S_{1} S_{3} S_{5} S_{4} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{5} S_{1} S_{3} S_{4} S_{3} S_{1} S_{2} S_{4} S_{2} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{6} \\<br />
S_{4} S_{3} S_{2} S_{5} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{3} S_{2} S_{2} S_{3} S_{4} S_{1} S_{1} S_{5} S_{7} S_{8} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{2} S_{5} S_{4} S_{4} S_{1} S_{1} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{5} S_{2} S_{3} S_{1} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{5} S_{2} S_{1} S_{4} S_{4} S_{2} S_{3} S_{3} S_{8} S_{8} S_{7} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{3} S_{3} S_{5} S_{5} S_{2} S_{1} S_{2} S_{4} S_{1} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{1} S_{3} S_{5} S_{2} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{3} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{6} \\<br />
S_{3} S_{2} S_{5} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{1} S_{5} S_{2} S_{2} S_{3} S_{3} S_{1} S_{5} S_{4} S_{4} S_{7} S_{8} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{3} S_{1} S_{4} S_{4} S_{2} S_{2} S_{1} S_{5} S_{3} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{6} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{4} S_{1} S_{5} S_{2} S_{1} S_{2} S_{3} S_{8} S_{7} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{4} S_{2} S_{2} S_{3} S_{5} S_{5} S_{1} S_{3} S_{1} S_{4} S_{7} S_{7} S_{8} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{2} S_{5} S_{3} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{5} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{4} S_{1} S_{3} S_{2} S_{5} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{1} S_{1} S_{3} S_{3} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{2} S_{5} S_{4} S_{1} S_{5} S_{2} S_{4} S_{4} S_{4} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{1} S_{2} S_{3} S_{4} S_{5} S_{5} S_{2} S_{5} S_{2} S_{5} S_{2} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{3} S_{4} S_{5} S_{4} S_{1} S_{2} S_{1} S_{4} S_{3} S_{1} S_{1} S_{7} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{7} S_{7} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{2} S_{4} S_{5} S_{4} S_{5} S_{4} S_{3} S_{4} S_{3} S_{2} S_{3} S_{1} S_{3} S_{5} S_{2} S_{1} S_{1} S_{5} S_{1} S_{2} S_{7} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{2} S_{1} S_{5} S_{4} S_{3} S_{8} S_{8} S_{7} \\<br />
S_{6} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{4} S_{3} S_{2} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{4} S_{5} S_{1} S_{5} S_{3} S_{4} S_{2} S_{5} S_{1} S_{2} S_{3} S_{3} S_{2} S_{1} S_{7} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} \\<br />
S_{1} S_{2} S_{1} S_{1} S_{4} S_{4} S_{2} S_{2} S_{5} S_{3} S_{4} S_{3} S_{3} S_{5} S_{5} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{1} S_{3} S_{5} S_{4} S_{2} S_{7} S_{7} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{1} S_{4} S_{5} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{2} S_{4} S_{5} S_{5} S_{4} S_{2} S_{1} S_{1} S_{3} S_{1} S_{3} S_{3} S_{2} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{10} \\<br />
S_{4} S_{4} S_{4} S_{2} S_{1} S_{1} S_{5} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{5} S_{3} S_{8} S_{8} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{4} S_{1} S_{2} S_{8} S_{7} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{8} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{7} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{3} S_{1} S_{4} S_{5} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{2} S_{5} S_{3} S_{5} S_{1} S_{4} S_{3} S_{4} S_{1} S_{2} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{4} S_{3} S_{2} S_{1} S_{5} S_{8} S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{7} S_{11} \\<br />
S_{9} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{2} S_{4} S_{3} S_{1} S_{5} S_{8} S_{7} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{1} S_{3} S_{3} S_{5} S_{5} S_{2} S_{4} S_{4} S_{1} S_{2} S_{8} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{3} S_{2} S_{1} S_{5} S_{4} S_{7} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{8} S_{7} \\<br />
S_{10} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{9} \\<br />
S_{7} \\<br />
S_{6} \\<br />
S_{1} S_{4} S_{3} S_{5} S_{2} \\<br />
S_{6} \\<br />
S_{2} S_{1} S_{3} S_{4} S_{5} \\<br />
S_{6} \\<br />
S_{7} \\<br />
S_{9} \\<br />
S_{11} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
8 & 6 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 8 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 8 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 8 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 6 & 8 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
16 & 16 & 16 & 16 & 16 & 48 & 20 & 28 & 8 & 12 & 3 \\<br />
8 & 8 & 8 & 8 & 8 & 20 & 12 & 10 & 5 & 6 & 2 \\<br />
8 & 8 & 8 & 8 & 8 & 28 & 10 & 20 & 2 & 7 & 0 \\<br />
4 & 4 & 4 & 4 & 4 & 8 & 5 & 2 & 4 & 2 & 2 \\<br />
4 & 4 & 4 & 4 & 4 & 12 & 6 & 7 & 2 & 4 & 0 \\<br />
2 & 2 & 2 & 2 & 2 & 3 & 2 & 0 & 2 & 0 & 2<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 4 & 1 & 3 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 1 & 0 & 1 & 0 & 1<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,31)&diff=1085M(32,51,31)2019-12-09T16:11:39Z<p>CesareGArdito: /* Projective indecomposable modules */</p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,32) - <math>B_0(k({\rm Aut}SL_2(32)))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k({\rm Aut}SL_2(32)))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{31}:C_5</math><br />
|k(B) = 16<br />
|l(B) = 11<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}({\rm Aut}SL_2(32)))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,30)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,31), then <math>B</math> is in [[M(32,51,23)]] or M(32,51,31).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{11}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{2} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{2} S_{2} S_{1} S_{4} S_{3} S_{5} S_{1} S_{3} S_{5} S_{4} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{5} S_{1} S_{3} S_{4} S_{3} S_{1} S_{2} S_{4} S_{2} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{6} \\<br />
S_{4} S_{3} S_{2} S_{5} S_{1} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{3} S_{2} S_{2} S_{3} S_{4} S_{1} S_{1} S_{5} S_{7} S_{8} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{2} S_{5} S_{4} S_{4} S_{1} S_{1} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{5} S_{2} S_{3} S_{1} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{6} \\<br />
S_{5} S_{4} S_{3} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{5} S_{2} S_{1} S_{4} S_{4} S_{2} S_{3} S_{3} S_{8} S_{8} S_{7} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{3} S_{3} S_{5} S_{5} S_{2} S_{1} S_{2} S_{4} S_{1} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{4} S_{1} S_{3} S_{5} S_{2} S_{7} S_{8} \\<br />
S_{6} \\<br />
S_{3} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{6} \\<br />
S_{3} S_{2} S_{5} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{1} S_{5} S_{2} S_{2} S_{3} S_{3} S_{1} S_{5} S_{4} S_{4} S_{7} S_{8} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{3} S_{1} S_{4} S_{4} S_{2} S_{2} S_{1} S_{5} S_{3} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{1} S_{2} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{6} \\<br />
S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{4} S_{4} S_{1} S_{5} S_{2} S_{1} S_{2} S_{3} S_{8} S_{7} S_{8} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\<br />
S_{4} S_{2} S_{2} S_{3} S_{5} S_{5} S_{1} S_{3} S_{1} S_{4} S_{7} S_{7} S_{8} S_{8} S_{8} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\<br />
S_{2} S_{5} S_{3} S_{4} S_{1} S_{8} S_{7} \\<br />
S_{6} \\<br />
S_{5} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{4} S_{1} S_{3} S_{2} S_{5} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{5} S_{1} S_{1} S_{3} S_{3} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{2} S_{5} S_{4} S_{1} S_{5} S_{2} S_{4} S_{4} S_{4} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{5} S_{3} S_{1} S_{2} S_{3} S_{4} S_{5} S_{5} S_{2} S_{5} S_{2} S_{5} S_{2} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{3} S_{4} S_{5} S_{4} S_{1} S_{2} S_{1} S_{4} S_{3} S_{1} S_{1} S_{7} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{7} S_{7} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{7} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{10} S_{9} S_{9} S_{10} \\<br />
S_{2} S_{4} S_{5} S_{4} S_{5} S_{4} S_{3} S_{4} S_{3} S_{2} S_{3} S_{1} S_{3} S_{5} S_{2} S_{1} S_{1} S_{5} S_{1} S_{2} S_{7} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{2} S_{1} S_{5} S_{4} S_{3} S_{8} S_{8} S_{7} \\<br />
S_{6} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{5} S_{1} S_{4} S_{3} S_{2} S_{8} S_{7} S_{7} S_{8} S_{11} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\<br />
S_{4} S_{4} S_{5} S_{1} S_{5} S_{3} S_{4} S_{2} S_{5} S_{1} S_{2} S_{3} S_{3} S_{2} S_{1} S_{7} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} \\<br />
S_{1} S_{2} S_{1} S_{1} S_{4} S_{4} S_{2} S_{2} S_{5} S_{3} S_{4} S_{3} S_{3} S_{5} S_{5} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\<br />
S_{1} S_{3} S_{5} S_{4} S_{2} S_{7} S_{7} S_{8} S_{8} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{1} S_{4} S_{5} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{4} S_{5} S_{2} S_{4} S_{5} S_{5} S_{4} S_{2} S_{1} S_{1} S_{3} S_{1} S_{3} S_{3} S_{2} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{7} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{10} \\<br />
S_{4} S_{4} S_{4} S_{2} S_{1} S_{1} S_{5} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{5} S_{3} S_{8} S_{8} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\<br />
S_{3} S_{5} S_{4} S_{1} S_{2} S_{8} S_{7} S_{7} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{10} \\<br />
S_{8} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{7} S_{11} \\<br />
S_{6} S_{10} S_{9} \\<br />
S_{3} S_{1} S_{4} S_{5} S_{2} S_{8} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{2} S_{5} S_{3} S_{5} S_{1} S_{4} S_{3} S_{4} S_{1} S_{2} S_{7} \\<br />
S_{6} S_{6} S_{6} \\<br />
S_{4} S_{3} S_{2} S_{1} S_{5} S_{8} S_{7} \\<br />
S_{6} S_{9} S_{10} \\<br />
S_{7} S_{11} \\<br />
S_{9} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{2} S_{4} S_{3} S_{1} S_{5} S_{8} S_{7} S_{7} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{1} S_{3} S_{3} S_{5} S_{5} S_{2} S_{4} S_{4} S_{1} S_{2} S_{8} S_{8} S_{8} \\<br />
S_{6} S_{6} S_{6} S_{6} S_{10} \\<br />
S_{3} S_{2} S_{1} S_{5} S_{4} S_{7} S_{7} S_{8} \\<br />
S_{6} S_{6} S_{9} \\<br />
S_{8} S_{7} \\<br />
S_{10} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{9} \\<br />
S_{7} \\<br />
S_{6} \\<br />
S_{1} S_{4} S_{3} S_{5} S_{2} \\<br />
S_{6} \\<br />
S_{2} S_{1} S_{3} S_{4} S_{5} \\<br />
S_{6} \\<br />
S_{7} \\<br />
S_{9} \\<br />
S_{11} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
8 & 6 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 8 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 8 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 8 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
6 & 6 & 6 & 6 & 8 & 16 & 8 & 8 & 4 & 4 & 2 \\<br />
16 & 16 & 16 & 16 & 16 & 48 & 20 & 28 & 8 & 12 & 3 \\<br />
8 & 8 & 8 & 8 & 8 & 20 & 12 & 10 & 5 & 6 & 2 \\<br />
8 & 8 & 8 & 8 & 8 & 28 & 10 & 20 & 2 & 7 & 0 \\<br />
4 & 4 & 4 & 4 & 4 & 8 & 5 & 2 & 4 & 2 & 2 \\<br />
4 & 4 & 4 & 4 & 4 & 12 & 6 & 7 & 2 & 4 & 0 \\<br />
2 & 2 & 2 & 2 & 2 & 3 & 2 & 0 & 2 & 0 & 2<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 4 & 1 & 3 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 1 & 0 & 1 & 0 & 1<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,34)&diff=1084M(32,51,34)2019-12-09T16:08:08Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,34) - <math>b_2(k((SL_2(8)\times (C_2)^2):3^{1+2}_+))</math> <br />
|image = &nbsp; <br />
|representative = <math>b_2(k((SL_2(8)\times (C_2)^2):3^{1+2}_+))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_7:C_3 \times C_3</math><br />
|k(B) = 16<br />
|l(B) = 7<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = <math>\left( \begin{array}{ccccccc}<br />
32 & 16 & 16 & 16 & 8 & 8 & 8 \\<br />
16 & 12 & 10 & 10 & 4 & 5 & 3 \\<br />
16 & 10 & 12 & 10 & 3 & 4 & 5 \\<br />
16 & 10 & 10 & 12 & 5 & 3 & 4 \\<br />
8 & 4 & 3 & 5 & 4 & 2 & 2 \\<br />
8 & 5 & 4 & 3 & 2 & 4 & 2 \\<br />
8 & 3 & 5 & 4 & 2 & 2 & 4<br />
\end{array} \right)</math><br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>b_2(\mathcal{O}((SL_2(8)\times (C_2)^2):3^{1+2}_+))</math><br />
|decomp = <math>\left( \begin{array}{ccccccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 1 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 \\<br />
3 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
3 & 2 & 2 & 2 & 1 & 1 & 1 <br />
\end{array}\right)</math><br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = No<br />
|k-derived =<br />
|O-derived-known? = No<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
This Morita equivalence class contains only non-principal blocks.<br />
<br />
It is unknown whether this class is derived equivalent to [[M(32,51,33)]]; if not, it forms its own derived equivalence class.<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
Any nonprincipal block with defect group <math>(C_2)^5</math> of <math>(SL_2(8)\times (C_2)^2):3^{1+2}_-</math>.<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,34), then <math>B</math> is in [[M(32,51,15)]], [[M(32,51,19)]] or M(32,51,34).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>b_2</math>-modules by <math>S_1,\dots, S_7 </math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{cccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{1} S_{1} S_{3} S_{4} S_{2} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{2} S_{6} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{3} S_{2} S_{2} S_{3} S_{4} S_{4} S_{4} S_{5} S_{7} S_{6} S_{5} S_{7} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{2} S_{3} S_{4} S_{3} S_{3} S_{2} S_{2} S_{4} S_{4} S_{3} S_{6} S_{7} S_{6} S_{5} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{4} S_{2} S_{3} S_{3} S_{4} S_{3} S_{2} S_{5} S_{5} S_{7} S_{6} S_{6} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{2} S_{3} S_{3} S_{4} S_{4} S_{2} S_{6} S_{7} S_{5} \\<br />
S_{1} S_{1} S_{3} S_{4} S_{2} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{1} S_{3} S_{4} S_{6} \\<br />
S_{1} S_{1} S_{3} S_{2} S_{4} S_{2} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{3} S_{4} S_{2} S_{2} S_{3} S_{4} S_{5} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{4} S_{3} S_{4} S_{2} S_{6} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{2} S_{3} S_{4} S_{3} S_{2} S_{4} S_{6} S_{5} \\<br />
S_{1} S_{1} S_{4} S_{2} S_{3} S_{2} S_{5} S_{7} \\<br />
S_{1} S_{3} S_{4} S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{1} S_{2} S_{4} S_{7} \\<br />
S_{1} S_{1} S_{3} S_{2} S_{3} S_{4} S_{5} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{3} S_{4} S_{4} S_{3} S_{2} S_{2} S_{7} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{4} S_{2} S_{3} S_{4} S_{2} S_{3} S_{7} S_{5} \\<br />
S_{1} S_{1} S_{1} S_{4} S_{3} S_{4} S_{3} S_{2} S_{2} S_{7} S_{6} \\<br />
S_{1} S_{1} S_{4} S_{3} S_{2} S_{3} S_{6} S_{5} \\<br />
S_{1} S_{4} S_{2} S_{7} \\<br />
S_{3} \\<br />
\end{array}<br />
\end{array}</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{1} S_{2} S_{3} S_{5} \\<br />
S_{1} S_{1} S_{2} S_{4} S_{3} S_{4} S_{7} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{4} S_{3} S_{2} S_{4} S_{3} S_{2} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{4} S_{3} S_{4} S_{2} S_{6} S_{5} \\<br />
S_{1} S_{1} S_{1} S_{2} S_{3} S_{4} S_{3} S_{2} S_{4} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{4} S_{3} S_{4} S_{2} S_{7} S_{6} \\<br />
S_{1} S_{2} S_{3} S_{5} \\<br />
S_{4} \\<br />
<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{4} S_{7} S_{6} \\<br />
S_{1} S_{3} S_{2} S_{5} \\<br />
S_{1} S_{1} S_{2} S_{4} \\<br />
S_{1} S_{1} S_{4} S_{3} \\<br />
S_{1} S_{1} S_{2} S_{4} \\<br />
S_{1} S_{2} S_{3} S_{5} \\<br />
S_{4} S_{7} S_{6} \\<br />
S_{5} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{2} S_{5} S_{7} \\<br />
S_{1} S_{4} S_{3} S_{6} \\<br />
S_{1} S_{1} S_{3} S_{2} \\<br />
S_{1} S_{1} S_{4} S_{2} \\<br />
S_{1} S_{1} S_{3} S_{2} \\<br />
S_{1} S_{4} S_{3} S_{6} \\<br />
S_{2} S_{7} S_{5} \\<br />
S_{6} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{3} S_{6} S_{5} \\<br />
S_{1} S_{4} S_{2} S_{7} \\<br />
S_{1} S_{1} S_{4} S_{3} \\<br />
S_{1} S_{1} S_{3} S_{2} \\<br />
S_{1} S_{1} S_{4} S_{3} \\<br />
S_{1} S_{2} S_{4} S_{7} \\<br />
S_{3} S_{5} S_{6} \\<br />
S_{7} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,34)&diff=1083M(32,51,34)2019-12-09T16:07:55Z<p>CesareGArdito: Created page with "{{blockbox |title = M(32,51,34) - <math>b_2(k((SL_2(8)\times (C_2)^2):3^{1+2}_+))</math> |image = &nbsp; |representative = <math>b_2(k((SL_2(8)\times (C_2)^2):3^{1+2}_+))</..."</p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,34) - <math>b_2(k((SL_2(8)\times (C_2)^2):3^{1+2}_+))</math> <br />
|image = &nbsp; <br />
|representative = <math>b_2(k((SL_2(8)\times (C_2)^2):3^{1+2}_+))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_7:C_3 \times C_3</math><br />
|k(B) = 16<br />
|l(B) = 7<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = <math>\left( \begin{array}{ccccccc}<br />
32 & 16 & 16 & 16 & 8 & 8 & 8 \\<br />
16 & 12 & 10 & 10 & 4 & 5 & 3 \\<br />
16 & 10 & 12 & 10 & 3 & 4 & 5 \\<br />
16 & 10 & 10 & 12 & 5 & 3 & 4 \\<br />
8 & 4 & 3 & 5 & 4 & 2 & 2 \\<br />
8 & 5 & 4 & 3 & 2 & 4 & 2 \\<br />
8 & 3 & 5 & 4 & 2 & 2 & 4<br />
\end{array} \right)</math><br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>b_2(\cO ((SL_2(8)\times (C_2)^2):3^{1+2}_+))</math><br />
|decomp = <math>\left( \begin{array}{ccccccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 1 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 \\<br />
3 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
3 & 2 & 2 & 2 & 1 & 1 & 1 <br />
\end{array}\right)</math><br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = No<br />
|k-derived =<br />
|O-derived-known? = No<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
This Morita equivalence class contains only non-principal blocks.<br />
<br />
It is unknown whether this class is derived equivalent to [[M(32,51,33)]]; if not, it forms its own derived equivalence class.<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
Any nonprincipal block with defect group <math>(C_2)^5</math> of <math>(SL_2(8)\times (C_2)^2):3^{1+2}_-</math>.<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,34), then <math>B</math> is in [[M(32,51,15)]], [[M(32,51,19)]] or M(32,51,34).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>b_2</math>-modules by <math>S_1,\dots, S_7 </math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{cccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{1} S_{1} S_{3} S_{4} S_{2} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{2} S_{6} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{3} S_{2} S_{2} S_{3} S_{4} S_{4} S_{4} S_{5} S_{7} S_{6} S_{5} S_{7} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{2} S_{3} S_{4} S_{3} S_{3} S_{2} S_{2} S_{4} S_{4} S_{3} S_{6} S_{7} S_{6} S_{5} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{4} S_{2} S_{3} S_{3} S_{4} S_{3} S_{2} S_{5} S_{5} S_{7} S_{6} S_{6} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{2} S_{3} S_{3} S_{4} S_{4} S_{2} S_{6} S_{7} S_{5} \\<br />
S_{1} S_{1} S_{3} S_{4} S_{2} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{1} S_{3} S_{4} S_{6} \\<br />
S_{1} S_{1} S_{3} S_{2} S_{4} S_{2} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{3} S_{4} S_{2} S_{2} S_{3} S_{4} S_{5} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{4} S_{3} S_{4} S_{2} S_{6} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{2} S_{3} S_{4} S_{3} S_{2} S_{4} S_{6} S_{5} \\<br />
S_{1} S_{1} S_{4} S_{2} S_{3} S_{2} S_{5} S_{7} \\<br />
S_{1} S_{3} S_{4} S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{1} S_{2} S_{4} S_{7} \\<br />
S_{1} S_{1} S_{3} S_{2} S_{3} S_{4} S_{5} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{3} S_{4} S_{4} S_{3} S_{2} S_{2} S_{7} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{4} S_{2} S_{3} S_{4} S_{2} S_{3} S_{7} S_{5} \\<br />
S_{1} S_{1} S_{1} S_{4} S_{3} S_{4} S_{3} S_{2} S_{2} S_{7} S_{6} \\<br />
S_{1} S_{1} S_{4} S_{3} S_{2} S_{3} S_{6} S_{5} \\<br />
S_{1} S_{4} S_{2} S_{7} \\<br />
S_{3} \\<br />
\end{array}<br />
\end{array}</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{1} S_{2} S_{3} S_{5} \\<br />
S_{1} S_{1} S_{2} S_{4} S_{3} S_{4} S_{7} S_{6} \\<br />
S_{1} S_{1} S_{1} S_{4} S_{3} S_{2} S_{4} S_{3} S_{2} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{4} S_{3} S_{4} S_{2} S_{6} S_{5} \\<br />
S_{1} S_{1} S_{1} S_{2} S_{3} S_{4} S_{3} S_{2} S_{4} S_{5} S_{7} \\<br />
S_{1} S_{1} S_{4} S_{3} S_{4} S_{2} S_{7} S_{6} \\<br />
S_{1} S_{2} S_{3} S_{5} \\<br />
S_{4} \\<br />
<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{4} S_{7} S_{6} \\<br />
S_{1} S_{3} S_{2} S_{5} \\<br />
S_{1} S_{1} S_{2} S_{4} \\<br />
S_{1} S_{1} S_{4} S_{3} \\<br />
S_{1} S_{1} S_{2} S_{4} \\<br />
S_{1} S_{2} S_{3} S_{5} \\<br />
S_{4} S_{7} S_{6} \\<br />
S_{5} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{2} S_{5} S_{7} \\<br />
S_{1} S_{4} S_{3} S_{6} \\<br />
S_{1} S_{1} S_{3} S_{2} \\<br />
S_{1} S_{1} S_{4} S_{2} \\<br />
S_{1} S_{1} S_{3} S_{2} \\<br />
S_{1} S_{4} S_{3} S_{6} \\<br />
S_{2} S_{7} S_{5} \\<br />
S_{6} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{3} S_{6} S_{5} \\<br />
S_{1} S_{4} S_{2} S_{7} \\<br />
S_{1} S_{1} S_{4} S_{3} \\<br />
S_{1} S_{1} S_{3} S_{2} \\<br />
S_{1} S_{1} S_{4} S_{3} \\<br />
S_{1} S_{2} S_{4} S_{7} \\<br />
S_{3} S_{5} S_{6} \\<br />
S_{7} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,11)&diff=1082M(32,51,11)2019-12-09T15:55:16Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,11) - <math>k(((C_2)^4 : C_{15}) \times C_2)</math> <br />
|image = &nbsp; <br />
|representative = <math>k(((C_2)^4 : C_{15}) \times C_2)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{15}</math>, ?<br />
|k(B) = 32<br />
|l(B) = 15<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O} (((C_2)^4 : C_{15}) \times C_2)</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,12)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is in this Morita equivalence class or in [[M(32,51,12)]], which is derived equivalent to this class. <br />
<br />
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,11), then <math>B</math> is in [[M(32,51,2)]], [[M(32,51,5)]], or M(32,51,11).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{15}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_1 \\<br />
S_1 S_{10} S_{11} S_9 S_4 \\<br />
S_{11} S_9 S_4 S_{10} S_{12} S_3 S_{14} S_{15} S_2 S_{13} \\<br />
S_3 S_{12} S_{14} S_2 S_{15} S_{13} S_5 S_6 S_8 S_7 \\<br />
S_6 S_5 S_7 S_8 S_1 \\<br />
S_1 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_2 \\<br />
S_{12} S_{14} S_5 S_7 S_2 \\<br />
S_{10} S_7 S_{14} S_6 S_8 S_4 S_1 S_5 S_{12} S_{15} \\<br />
S_{15} S_6 S_1 S_8 S_4 S_{10} S_3 S_9 S_{13} S_{11} \\<br />
S_9 S_{13} S_3 S_{11} S_2 \\<br />
S_2 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_3 \\<br />
S_3 S_7 S_2 S_6 S_{11} \\<br />
S_{12} S_7 S_{11} S_5 S_2 S_4 S_6 S_1 S_{14} S_{13} \\<br />
S_{14} S_4 S_1 S_5 S_{12} S_{13} S_{15} S_9 S_8 S_{10} \\<br />
S_9 S_8 S_{10} S_{15} S_3 \\<br />
S_3 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_4 \\<br />
S_{13} S_9 S_{14} S_{15} S_4 \\<br />
S_{14} S_2 S_8 S_6 S_3 S_{10} S_{15} S_5 S_9 S_{13} \\<br />
S_8 S_{10} S_1 S_5 S_2 S_7 S_3 S_6 S_{12} S_{11} \\<br />
S_7 S_1 S_{12} S_{11} S_4 \\<br />
S_4 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_5 \\<br />
S_1 S_{10} S_{12} S_5 S_8 \\<br />
S_4 S_7 S_1 S_{10} S_8 S_9 S_3 S_{15} S_{12} S_{11} \\<br />
S_{14} S_{11} S_{13} S_2 S_4 S_7 S_{15} S_3 S_9 S_6 \\<br />
S_{13} S_{14} S_6 S_5 S_2 \\<br />
S_5 \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_6 \\<br />
S_1 S_5 S_{11} S_6 S_{13} \\<br />
S_{12} S_{11} S_{10} S_1 S_5 S_2 S_8 S_9 S_4 S_{13} \\<br />
S_9 S_3 S_{10} S_{14} S_{15} S_8 S_7 S_2 S_4 S_{12} \\<br />
S_7 S_{14} S_{15} S_3 S_6 \\<br />
S_6 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_7 \\<br />
S_4 S_6 S_{14} S_1 S_7 \\<br />
S_5 S_9 S_{13} S_{14} S_4 S_6 S_{11} S_{15} S_1 S_{10} \\<br />
S_3 S_{13} S_8 S_2 S_9 S_{12} S_{15} S_5 S_{11} S_{10} \\<br />
S_{12} S_2 S_7 S_3 S_8 \\<br />
S_7 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_8 \\<br />
S_8 S_1 S_3 S_7 S_9 \\<br />
S_9 S_{14} S_1 S_4 S_{10} S_6 S_2 S_{11} S_7 S_3 \\<br />
S_{14} S_{15} S_6 S_2 S_4 S_{11} S_{10} S_{13} S_{12} S_5 \\<br />
S_{15} S_5 S_{13} S_8 S_{12} \\<br />
S_8 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_9 \\<br />
S_3 S_{14} S_{10} S_2 S_9 \\<br />
S_{10} S_{15} S_2 S_7 S_6 S_{14} S_{12} S_{11} S_5 S_3 \\<br />
S_4 S_6 S_5 S_{12} S_{13} S_{11} S_7 S_1 S_{15} S_8 \\<br />
S_1 S_{13} S_8 S_4 S_9 \\<br />
S_9 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{15} S_{12} S_3 S_{10} S_{11} \\<br />
S_4 S_2 S_{13} S_8 S_7 S_6 S_3 S_{15} S_{11} S_{12} \\<br />
S_9 S_5 S_7 S_{14} S_6 S_{13} S_2 S_8 S_4 S_1 \\<br />
S_{14} S_5 S_9 S_1 S_{10} \\<br />
S_{10} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{13} S_{11} S_2 S_{12} S_4 \\<br />
S_8 S_4 S_9 S_7 S_{12} S_{15} S_2 S_{13} S_{14} S_5 \\<br />
S_1 S_{14} S_8 S_3 S_5 S_9 S_6 S_{15} S_{10} S_7 \\<br />
S_1 S_{10} S_6 S_3 S_{11} \\<br />
S_{11} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{12} \\<br />
S_{15} S_4 S_{12} S_7 S_8 \\<br />
S_6 S_3 S_4 S_9 S_1 S_7 S_{14} S_{13} S_{15} S_8 \\<br />
S_2 S_{13} S_3 S_1 S_9 S_6 S_{10} S_{11} S_{14} S_5 \\<br />
S_{11} S_5 S_2 S_{10} S_{12} \\<br />
S_{12} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{13} \\<br />
S_8 S_5 S_{13} S_9 S_2 \\<br />
S_3 S_{12} S_7 S_1 S_5 S_{10} S_9 S_8 S_2 S_{14} \\<br />
S_{12} S_7 S_3 S_{10} S_{14} S_1 S_{11} S_{15} S_6 S_4 \\<br />
S_{11} S_4 S_{15} S_6 S_{13} \\<br />
S_{13} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{14} \\<br />
S_6 S_{10} S_{14} S_{15} S_5 \\<br />
S_{15} S_5 S_3 S_{12} S_{13} S_{11} S_1 S_{10} S_8 S_6 \\<br />
S_2 S_{12} S_8 S_{13} S_3 S_1 S_{11} S_4 S_9 S_7 \\<br />
S_7 S_4 S_9 S_2 S_{14} \\<br />
S_{14} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{15} \\<br />
S_{15} S_6 S_8 S_{13} S_3 \\<br />
S_2 S_8 S_{13} S_3 S_5 S_1 S_6 S_9 S_{11} S_7 \\<br />
S_2 S_{11} S_7 S_5 S_1 S_9 S_{10} S_4 S_{12} S_{14} \\<br />
S_{14} S_{12} S_{10} S_4 S_{15} \\<br />
S_{15} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 <br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,12)&diff=1081M(32,51,12)2019-12-09T15:55:01Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,12) - <math>B_0(k(SL_2(16) \times C_2))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k(SL_2(16) \times C_2))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{15}</math>, ?<br />
|k(B) = 32<br />
|l(B) = 15<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}(SL_2(16) \times C_2))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,11)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is in this Morita equivalence class or in [[M(32,51,11)]], which is derived equivalent to this class. <br />
<br />
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,12), then <math>B</math> is also in M(32,51,12).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
32 & 16 & 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 \\<br />
16 & 16 & 8 & 8 & 8 & 4 & 8 & 4 & 4 & 8 & 0 & 0 & 2 & 4 & 0 \\<br />
16 & 8 & 16 & 8 & 8 & 8 & 4 & 0 & 4 & 4 & 8 & 0 & 0 & 2 & 4 \\<br />
16 & 8 & 8 & 16 & 8 & 4 & 8 & 8 & 0 & 4 & 4 & 4 & 0 & 0 & 2 \\<br />
16 & 8 & 8 & 8 & 16 & 8 & 4 & 4 & 8 & 0 & 4 & 2 & 4 & 0 & 0 \\<br />
8 & 4 & 8 & 4 & 8 & 8 & 2 & 0 & 4 & 0 & 4 & 0 & 0 & 0 & 0 \\<br />
8 & 8 & 4 & 8 & 4 & 2 & 8 & 4 & 0 & 4 & 0 & 0 & 0 & 0 & 0 \\<br />
8 & 4 & 0 & 8 & 4 & 0 & 4 & 8 & 0 & 2 & 0 & 4 & 0 & 0 & 0 \\<br />
8 & 4 & 4 & 0 & 8 & 4 & 0 & 0 & 8 & 0 & 2 & 0 & 4 & 0 & 0 \\<br />
8 & 8 & 4 & 4 & 0 & 0 & 4 & 2 & 0 & 8 & 0 & 0 & 0 & 4 & 0 \\<br />
8 & 0 & 8 & 4 & 4 & 4 & 0 & 0 & 2 & 0 & 8 & 0 & 0 & 0 & 4 \\<br />
4 & 0 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\<br />
4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 \\<br />
4 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 \\<br />
4 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 \\<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&diff=1080M(32,51,5)2019-12-09T15:54:32Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,5) - <math>k((C_2)^4 : C_5) \times C_2)</math> <br />
|image = &nbsp; <br />
|representative = <math>k((C_2)^4 : C_5) \times C_2)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_5</math>, ?<br />
|k(B) = 16<br />
|l(B) = 5<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = <math>\left( \begin{array}{ccc}<br />
8 & 6 & 6 & 6 & 6 \\<br />
6 & 8 & 6 & 6 & 6 \\<br />
6 & 6 & 8 & 6 & 6 \\<br />
6 & 6 & 6 & 8 & 6 \\<br />
6 & 6 & 6 & 6 & 8<br />
\end{array} \right)</math><br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O} ((C_2)^4 : C_5) \times C_2)</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = Forms a derived equivalence class<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is in this Morita equivalence class.<br />
<br />
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,5), then <math>B</math> is in [[M(32,51,1)]], M(32,51,5), or [[M(32,51,11)]].<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, S_2, S_3, S_4, S_5</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_1 \\<br />
S_1 S_5 S_2 S_4 S_3 \\<br />
S_3 S_5 S_4 S_2 S_1 S_4 S_3 S_1 S_5 S_2 \\<br />
S_2 S_2 S_3 S_5 S_4 S_1 S_3 S_5 S_1 S_4 \\<br />
S_4 S_5 S_2 S_3 S_1 \\<br />
S_1 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_2 \\<br />
S_4 S_2 S_1 S_5 S_3 \\<br />
S_5 S_1 S_3 S_1 S_4 S_4 S_2 S_2 S_5 S_3 \\<br />
S_2 S_3 S_1 S_5 S_4 S_2 S_3 S_4 S_5 S_1 \\<br />
S_3 S_4 S_1 S_5 S_2 \\<br />
S_2 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_3 \\<br />
S_5 S_4 S_2 S_3 S_1 \\<br />
S_4 S_2 S_3 S_3 S_4 S_1 S_2 S_1 S_5 S_5 \\<br />
S_4 S_2 S_3 S_2 S_1 S_1 S_4 S_5 S_5 S_3 \\<br />
S_5 S_2 S_4 S_1 S_3 \\<br />
S_3 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_4 \\<br />
S_5 S_1 S_2 S_3 S_4 \\<br />
S_1 S_3 S_2 S_2 S_3 S_5 S_5 S_1 S_4 S_4 \\<br />
S_5 S_3 S_4 S_4 S_3 S_2 S_5 S_2 S_1 S_1 \\<br />
S_3 S_5 S_2 S_1 S_4 \\<br />
S_4 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_5 \\<br />
S_2 S_4 S_5 S_1 S_3 \\<br />
S_5 S_3 S_5 S_2 S_4 S_2 S_3 S_4 S_1 S_1 \\<br />
S_2 S_4 S_1 S_5 S_2 S_5 S_1 S_4 S_3 S_3 \\<br />
S_2 S_3 S_1 S_4 S_5 \\<br />
S_5 \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 <br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&diff=1079M(32,51,5)2019-12-09T15:53:43Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,5) - <math>k((C_2)^4 : C_5) \times C_2)</math> <br />
|image = &nbsp; <br />
|representative = <math>k((C_2)^4 : C_5) \times C_2)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_5</math>, ?<br />
|k(B) = 16<br />
|l(B) = 5<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = <math>\left( \begin{array}{ccc}<br />
8 & 6 & 6 & 6 & 6 \\<br />
6 & 8 & 6 & 6 & 6 \\<br />
6 & 6 & 8 & 6 & 6 \\<br />
6 & 6 & 6 & 8 & 6 \\<br />
6 & 6 & 6 & 6 & 8<br />
\end{array} \right)</math><br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O} ((C_2)^4 : C_5) \times C_2)</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = Forms a derived equivalence class<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to M(32,51,5). <br />
<br />
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,5), then <math>B</math> is in [[M(32,51,1)]], M(32,51,5), or [[M(32,51,11)]].<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, S_2, S_3, S_4, S_5</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_1 \\<br />
S_1 S_5 S_2 S_4 S_3 \\<br />
S_3 S_5 S_4 S_2 S_1 S_4 S_3 S_1 S_5 S_2 \\<br />
S_2 S_2 S_3 S_5 S_4 S_1 S_3 S_5 S_1 S_4 \\<br />
S_4 S_5 S_2 S_3 S_1 \\<br />
S_1 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_2 \\<br />
S_4 S_2 S_1 S_5 S_3 \\<br />
S_5 S_1 S_3 S_1 S_4 S_4 S_2 S_2 S_5 S_3 \\<br />
S_2 S_3 S_1 S_5 S_4 S_2 S_3 S_4 S_5 S_1 \\<br />
S_3 S_4 S_1 S_5 S_2 \\<br />
S_2 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_3 \\<br />
S_5 S_4 S_2 S_3 S_1 \\<br />
S_4 S_2 S_3 S_3 S_4 S_1 S_2 S_1 S_5 S_5 \\<br />
S_4 S_2 S_3 S_2 S_1 S_1 S_4 S_5 S_5 S_3 \\<br />
S_5 S_2 S_4 S_1 S_3 \\<br />
S_3 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_4 \\<br />
S_5 S_1 S_2 S_3 S_4 \\<br />
S_1 S_3 S_2 S_2 S_3 S_5 S_5 S_1 S_4 S_4 \\<br />
S_5 S_3 S_4 S_4 S_3 S_2 S_5 S_2 S_1 S_1 \\<br />
S_3 S_5 S_2 S_1 S_4 \\<br />
S_4 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_5 \\<br />
S_2 S_4 S_5 S_1 S_3 \\<br />
S_5 S_3 S_5 S_2 S_4 S_2 S_3 S_4 S_1 S_1 \\<br />
S_2 S_4 S_1 S_5 S_2 S_5 S_1 S_4 S_3 S_3 \\<br />
S_2 S_3 S_1 S_4 S_5 \\<br />
S_5 \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 <br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,11)&diff=1078M(32,51,11)2019-12-09T15:53:28Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,11) - <math>k(((C_2)^4 : C_{15}) \times C_2)</math> <br />
|image = &nbsp; <br />
|representative = <math>k(((C_2)^4 : C_{15}) \times C_2)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{15}</math>, ?<br />
|k(B) = 32<br />
|l(B) = 15<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O} (((C_2)^4 : C_{15}) \times C_2)</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,12)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is Morita equivalent to M(32,51,11) or [[M(32,51,12)]], and the latter are derived equivalent. <br />
<br />
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,11), then <math>B</math> is in [[M(32,51,2)]], [[M(32,51,5)]], or M(32,51,11).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{15}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_1 \\<br />
S_1 S_{10} S_{11} S_9 S_4 \\<br />
S_{11} S_9 S_4 S_{10} S_{12} S_3 S_{14} S_{15} S_2 S_{13} \\<br />
S_3 S_{12} S_{14} S_2 S_{15} S_{13} S_5 S_6 S_8 S_7 \\<br />
S_6 S_5 S_7 S_8 S_1 \\<br />
S_1 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_2 \\<br />
S_{12} S_{14} S_5 S_7 S_2 \\<br />
S_{10} S_7 S_{14} S_6 S_8 S_4 S_1 S_5 S_{12} S_{15} \\<br />
S_{15} S_6 S_1 S_8 S_4 S_{10} S_3 S_9 S_{13} S_{11} \\<br />
S_9 S_{13} S_3 S_{11} S_2 \\<br />
S_2 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_3 \\<br />
S_3 S_7 S_2 S_6 S_{11} \\<br />
S_{12} S_7 S_{11} S_5 S_2 S_4 S_6 S_1 S_{14} S_{13} \\<br />
S_{14} S_4 S_1 S_5 S_{12} S_{13} S_{15} S_9 S_8 S_{10} \\<br />
S_9 S_8 S_{10} S_{15} S_3 \\<br />
S_3 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_4 \\<br />
S_{13} S_9 S_{14} S_{15} S_4 \\<br />
S_{14} S_2 S_8 S_6 S_3 S_{10} S_{15} S_5 S_9 S_{13} \\<br />
S_8 S_{10} S_1 S_5 S_2 S_7 S_3 S_6 S_{12} S_{11} \\<br />
S_7 S_1 S_{12} S_{11} S_4 \\<br />
S_4 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_5 \\<br />
S_1 S_{10} S_{12} S_5 S_8 \\<br />
S_4 S_7 S_1 S_{10} S_8 S_9 S_3 S_{15} S_{12} S_{11} \\<br />
S_{14} S_{11} S_{13} S_2 S_4 S_7 S_{15} S_3 S_9 S_6 \\<br />
S_{13} S_{14} S_6 S_5 S_2 \\<br />
S_5 \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_6 \\<br />
S_1 S_5 S_{11} S_6 S_{13} \\<br />
S_{12} S_{11} S_{10} S_1 S_5 S_2 S_8 S_9 S_4 S_{13} \\<br />
S_9 S_3 S_{10} S_{14} S_{15} S_8 S_7 S_2 S_4 S_{12} \\<br />
S_7 S_{14} S_{15} S_3 S_6 \\<br />
S_6 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_7 \\<br />
S_4 S_6 S_{14} S_1 S_7 \\<br />
S_5 S_9 S_{13} S_{14} S_4 S_6 S_{11} S_{15} S_1 S_{10} \\<br />
S_3 S_{13} S_8 S_2 S_9 S_{12} S_{15} S_5 S_{11} S_{10} \\<br />
S_{12} S_2 S_7 S_3 S_8 \\<br />
S_7 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_8 \\<br />
S_8 S_1 S_3 S_7 S_9 \\<br />
S_9 S_{14} S_1 S_4 S_{10} S_6 S_2 S_{11} S_7 S_3 \\<br />
S_{14} S_{15} S_6 S_2 S_4 S_{11} S_{10} S_{13} S_{12} S_5 \\<br />
S_{15} S_5 S_{13} S_8 S_{12} \\<br />
S_8 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_9 \\<br />
S_3 S_{14} S_{10} S_2 S_9 \\<br />
S_{10} S_{15} S_2 S_7 S_6 S_{14} S_{12} S_{11} S_5 S_3 \\<br />
S_4 S_6 S_5 S_{12} S_{13} S_{11} S_7 S_1 S_{15} S_8 \\<br />
S_1 S_{13} S_8 S_4 S_9 \\<br />
S_9 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{15} S_{12} S_3 S_{10} S_{11} \\<br />
S_4 S_2 S_{13} S_8 S_7 S_6 S_3 S_{15} S_{11} S_{12} \\<br />
S_9 S_5 S_7 S_{14} S_6 S_{13} S_2 S_8 S_4 S_1 \\<br />
S_{14} S_5 S_9 S_1 S_{10} \\<br />
S_{10} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{13} S_{11} S_2 S_{12} S_4 \\<br />
S_8 S_4 S_9 S_7 S_{12} S_{15} S_2 S_{13} S_{14} S_5 \\<br />
S_1 S_{14} S_8 S_3 S_5 S_9 S_6 S_{15} S_{10} S_7 \\<br />
S_1 S_{10} S_6 S_3 S_{11} \\<br />
S_{11} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{12} \\<br />
S_{15} S_4 S_{12} S_7 S_8 \\<br />
S_6 S_3 S_4 S_9 S_1 S_7 S_{14} S_{13} S_{15} S_8 \\<br />
S_2 S_{13} S_3 S_1 S_9 S_6 S_{10} S_{11} S_{14} S_5 \\<br />
S_{11} S_5 S_2 S_{10} S_{12} \\<br />
S_{12} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{13} \\<br />
S_8 S_5 S_{13} S_9 S_2 \\<br />
S_3 S_{12} S_7 S_1 S_5 S_{10} S_9 S_8 S_2 S_{14} \\<br />
S_{12} S_7 S_3 S_{10} S_{14} S_1 S_{11} S_{15} S_6 S_4 \\<br />
S_{11} S_4 S_{15} S_6 S_{13} \\<br />
S_{13} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{14} \\<br />
S_6 S_{10} S_{14} S_{15} S_5 \\<br />
S_{15} S_5 S_3 S_{12} S_{13} S_{11} S_1 S_{10} S_8 S_6 \\<br />
S_2 S_{12} S_8 S_{13} S_3 S_1 S_{11} S_4 S_9 S_7 \\<br />
S_7 S_4 S_9 S_2 S_{14} \\<br />
S_{14} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{15} \\<br />
S_{15} S_6 S_8 S_{13} S_3 \\<br />
S_2 S_8 S_{13} S_3 S_5 S_1 S_6 S_9 S_{11} S_7 \\<br />
S_2 S_{11} S_7 S_5 S_1 S_9 S_{10} S_4 S_{12} S_{14} \\<br />
S_{14} S_{12} S_{10} S_4 S_{15} \\<br />
S_{15} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 <br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,12)&diff=1077M(32,51,12)2019-12-09T15:53:18Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,12) - <math>B_0(k(SL_2(16) \times C_2))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k(SL_2(16) \times C_2))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{15}</math>, ?<br />
|k(B) = 32<br />
|l(B) = 15<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}(SL_2(16) \times C_2))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,11)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is Morita equivalent to [[M(32,51,11)]] or M(32,51,12), and the latter are derived equivalent. <br />
<br />
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,12), then <math>B</math> is also in M(32,51,12).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
32 & 16 & 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 \\<br />
16 & 16 & 8 & 8 & 8 & 4 & 8 & 4 & 4 & 8 & 0 & 0 & 2 & 4 & 0 \\<br />
16 & 8 & 16 & 8 & 8 & 8 & 4 & 0 & 4 & 4 & 8 & 0 & 0 & 2 & 4 \\<br />
16 & 8 & 8 & 16 & 8 & 4 & 8 & 8 & 0 & 4 & 4 & 4 & 0 & 0 & 2 \\<br />
16 & 8 & 8 & 8 & 16 & 8 & 4 & 4 & 8 & 0 & 4 & 2 & 4 & 0 & 0 \\<br />
8 & 4 & 8 & 4 & 8 & 8 & 2 & 0 & 4 & 0 & 4 & 0 & 0 & 0 & 0 \\<br />
8 & 8 & 4 & 8 & 4 & 2 & 8 & 4 & 0 & 4 & 0 & 0 & 0 & 0 & 0 \\<br />
8 & 4 & 0 & 8 & 4 & 0 & 4 & 8 & 0 & 2 & 0 & 4 & 0 & 0 & 0 \\<br />
8 & 4 & 4 & 0 & 8 & 4 & 0 & 0 & 8 & 0 & 2 & 0 & 4 & 0 & 0 \\<br />
8 & 8 & 4 & 4 & 0 & 0 & 4 & 2 & 0 & 8 & 0 & 0 & 0 & 4 & 0 \\<br />
8 & 0 & 8 & 4 & 4 & 4 & 0 & 0 & 2 & 0 & 8 & 0 & 0 & 0 & 4 \\<br />
4 & 0 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\<br />
4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 \\<br />
4 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 \\<br />
4 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 \\<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,12)&diff=1076M(32,51,12)2019-12-09T15:53:01Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,12) - <math>B_0(k(SL_2(16) \times C_2))</math> <br />
|image = &nbsp; <br />
|representative = <math>B_0(k(SL_2(16) \times C_2))</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{15}</math><br />
|k(B) = 32<br />
|l(B) = 15<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>B_0(\mathcal{O}(SL_2(16) \times C_2))</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,11)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is Morita equivalent to [[M(32,51,11)]] or M(32,51,12), and the latter are derived equivalent. <br />
<br />
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,12), then <math>B</math> is also in M(32,51,12).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
32 & 16 & 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 \\<br />
16 & 16 & 8 & 8 & 8 & 4 & 8 & 4 & 4 & 8 & 0 & 0 & 2 & 4 & 0 \\<br />
16 & 8 & 16 & 8 & 8 & 8 & 4 & 0 & 4 & 4 & 8 & 0 & 0 & 2 & 4 \\<br />
16 & 8 & 8 & 16 & 8 & 4 & 8 & 8 & 0 & 4 & 4 & 4 & 0 & 0 & 2 \\<br />
16 & 8 & 8 & 8 & 16 & 8 & 4 & 4 & 8 & 0 & 4 & 2 & 4 & 0 & 0 \\<br />
8 & 4 & 8 & 4 & 8 & 8 & 2 & 0 & 4 & 0 & 4 & 0 & 0 & 0 & 0 \\<br />
8 & 8 & 4 & 8 & 4 & 2 & 8 & 4 & 0 & 4 & 0 & 0 & 0 & 0 & 0 \\<br />
8 & 4 & 0 & 8 & 4 & 0 & 4 & 8 & 0 & 2 & 0 & 4 & 0 & 0 & 0 \\<br />
8 & 4 & 4 & 0 & 8 & 4 & 0 & 0 & 8 & 0 & 2 & 0 & 4 & 0 & 0 \\<br />
8 & 8 & 4 & 4 & 0 & 0 & 4 & 2 & 0 & 8 & 0 & 0 & 0 & 4 & 0 \\<br />
8 & 0 & 8 & 4 & 4 & 4 & 0 & 0 & 2 & 0 & 8 & 0 & 0 & 0 & 4 \\<br />
4 & 0 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\<br />
4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 \\<br />
4 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 \\<br />
4 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 \\<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\<br />
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\<br />
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,11)&diff=1075M(32,51,11)2019-12-09T15:52:51Z<p>CesareGArdito: </p>
<hr />
<div>{{blockbox<br />
|title = M(32,51,11) - <math>k(((C_2)^4 : C_{15}) \times C_2)</math> <br />
|image = &nbsp; <br />
|representative = <math>k(((C_2)^4 : C_{15}) \times C_2)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{15}</math><br />
|k(B) = 32<br />
|l(B) = 15<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O} (((C_2)^4 : C_{15}) \times C_2)</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,12)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_{15}</math> is Morita equivalent to M(32,51,11) or [[M(32,51,12)]], and the latter are derived equivalent. <br />
<br />
It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3 \times C_3</math> (with action as in [[M(32,51,24)]]).<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
Let <math>N \triangleleft G</math> with prime <math>p'</math>-index and let <math>B</math> be a block of <math>\mathcal{O} G</math> covering a block <math>b</math> of <math>\mathcal{O} N</math>.<br />
<br />
If <math>b</math> is in M(32,51,11), then <math>B</math> is in [[M(32,51,2)]], [[M(32,51,5)]], or M(32,51,11).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{15}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_1 \\<br />
S_1 S_{10} S_{11} S_9 S_4 \\<br />
S_{11} S_9 S_4 S_{10} S_{12} S_3 S_{14} S_{15} S_2 S_{13} \\<br />
S_3 S_{12} S_{14} S_2 S_{15} S_{13} S_5 S_6 S_8 S_7 \\<br />
S_6 S_5 S_7 S_8 S_1 \\<br />
S_1 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_2 \\<br />
S_{12} S_{14} S_5 S_7 S_2 \\<br />
S_{10} S_7 S_{14} S_6 S_8 S_4 S_1 S_5 S_{12} S_{15} \\<br />
S_{15} S_6 S_1 S_8 S_4 S_{10} S_3 S_9 S_{13} S_{11} \\<br />
S_9 S_{13} S_3 S_{11} S_2 \\<br />
S_2 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_3 \\<br />
S_3 S_7 S_2 S_6 S_{11} \\<br />
S_{12} S_7 S_{11} S_5 S_2 S_4 S_6 S_1 S_{14} S_{13} \\<br />
S_{14} S_4 S_1 S_5 S_{12} S_{13} S_{15} S_9 S_8 S_{10} \\<br />
S_9 S_8 S_{10} S_{15} S_3 \\<br />
S_3 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_4 \\<br />
S_{13} S_9 S_{14} S_{15} S_4 \\<br />
S_{14} S_2 S_8 S_6 S_3 S_{10} S_{15} S_5 S_9 S_{13} \\<br />
S_8 S_{10} S_1 S_5 S_2 S_7 S_3 S_6 S_{12} S_{11} \\<br />
S_7 S_1 S_{12} S_{11} S_4 \\<br />
S_4 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_5 \\<br />
S_1 S_{10} S_{12} S_5 S_8 \\<br />
S_4 S_7 S_1 S_{10} S_8 S_9 S_3 S_{15} S_{12} S_{11} \\<br />
S_{14} S_{11} S_{13} S_2 S_4 S_7 S_{15} S_3 S_9 S_6 \\<br />
S_{13} S_{14} S_6 S_5 S_2 \\<br />
S_5 \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_6 \\<br />
S_1 S_5 S_{11} S_6 S_{13} \\<br />
S_{12} S_{11} S_{10} S_1 S_5 S_2 S_8 S_9 S_4 S_{13} \\<br />
S_9 S_3 S_{10} S_{14} S_{15} S_8 S_7 S_2 S_4 S_{12} \\<br />
S_7 S_{14} S_{15} S_3 S_6 \\<br />
S_6 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_7 \\<br />
S_4 S_6 S_{14} S_1 S_7 \\<br />
S_5 S_9 S_{13} S_{14} S_4 S_6 S_{11} S_{15} S_1 S_{10} \\<br />
S_3 S_{13} S_8 S_2 S_9 S_{12} S_{15} S_5 S_{11} S_{10} \\<br />
S_{12} S_2 S_7 S_3 S_8 \\<br />
S_7 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_8 \\<br />
S_8 S_1 S_3 S_7 S_9 \\<br />
S_9 S_{14} S_1 S_4 S_{10} S_6 S_2 S_{11} S_7 S_3 \\<br />
S_{14} S_{15} S_6 S_2 S_4 S_{11} S_{10} S_{13} S_{12} S_5 \\<br />
S_{15} S_5 S_{13} S_8 S_{12} \\<br />
S_8 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_9 \\<br />
S_3 S_{14} S_{10} S_2 S_9 \\<br />
S_{10} S_{15} S_2 S_7 S_6 S_{14} S_{12} S_{11} S_5 S_3 \\<br />
S_4 S_6 S_5 S_{12} S_{13} S_{11} S_7 S_1 S_{15} S_8 \\<br />
S_1 S_{13} S_8 S_4 S_9 \\<br />
S_9 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{15} S_{12} S_3 S_{10} S_{11} \\<br />
S_4 S_2 S_{13} S_8 S_7 S_6 S_3 S_{15} S_{11} S_{12} \\<br />
S_9 S_5 S_7 S_{14} S_6 S_{13} S_2 S_8 S_4 S_1 \\<br />
S_{14} S_5 S_9 S_1 S_{10} \\<br />
S_{10} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{13} S_{11} S_2 S_{12} S_4 \\<br />
S_8 S_4 S_9 S_7 S_{12} S_{15} S_2 S_{13} S_{14} S_5 \\<br />
S_1 S_{14} S_8 S_3 S_5 S_9 S_6 S_{15} S_{10} S_7 \\<br />
S_1 S_{10} S_6 S_3 S_{11} \\<br />
S_{11} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{12} \\<br />
S_{15} S_4 S_{12} S_7 S_8 \\<br />
S_6 S_3 S_4 S_9 S_1 S_7 S_{14} S_{13} S_{15} S_8 \\<br />
S_2 S_{13} S_3 S_1 S_9 S_6 S_{10} S_{11} S_{14} S_5 \\<br />
S_{11} S_5 S_2 S_{10} S_{12} \\<br />
S_{12} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{13} \\<br />
S_8 S_5 S_{13} S_9 S_2 \\<br />
S_3 S_{12} S_7 S_1 S_5 S_{10} S_9 S_8 S_2 S_{14} \\<br />
S_{12} S_7 S_3 S_{10} S_{14} S_1 S_{11} S_{15} S_6 S_4 \\<br />
S_{11} S_4 S_{15} S_6 S_{13} \\<br />
S_{13} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{14} \\<br />
S_6 S_{10} S_{14} S_{15} S_5 \\<br />
S_{15} S_5 S_3 S_{12} S_{13} S_{11} S_1 S_{10} S_8 S_6 \\<br />
S_2 S_{12} S_8 S_{13} S_3 S_1 S_{11} S_4 S_9 S_7 \\<br />
S_7 S_4 S_9 S_2 S_{14} \\<br />
S_{14} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{15} \\<br />
S_{15} S_6 S_8 S_{13} S_3 \\<br />
S_2 S_8 S_{13} S_3 S_5 S_1 S_6 S_9 S_{11} S_7 \\<br />
S_2 S_{11} S_7 S_5 S_1 S_9 S_{10} S_4 S_{12} S_{14} \\<br />
S_{14} S_{12} S_{10} S_4 S_{15} \\<br />
S_{15} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccc}<br />
4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\<br />
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 <br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArdito