Difference between revisions of "M(4,2,1)"

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{{blockbox
 
{{blockbox
 
|title = M(4,2,1) - <math>k(C_2 \times C_2)</math>  
 
|title = M(4,2,1) - <math>k(C_2 \times C_2)</math>  
|image =  
+
|image = M(4,2,1)quiver.png
 
|representative =  <math>k(C_2 \times C_2)</math>
 
|representative =  <math>k(C_2 \times C_2)</math>
 
|defect = <math>C_2 \times C_2</math>
 
|defect = <math>C_2 \times C_2</math>
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|O-morita-frob = 1
 
|O-morita-frob = 1
 
|Pic-O = <math>\mathcal{L}(B)=S_4</math>
 
|Pic-O = <math>\mathcal{L}(B)=S_4</math>
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|PIgroup = <math>S_4 \times C_2</math>
 
|source? = Yes
 
|source? = Yes
 
|sourcereps = <math>k(C_2 \times C_2)</math>
 
|sourcereps = <math>k(C_2 \times C_2)</math>
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|k-derived = Forms a derived equivalence class
 
|k-derived = Forms a derived equivalence class
 
|O-derived-known? = Yes
 
|O-derived-known? = Yes
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|coveringblocks = M(4,2,1), [[M(4,2,3)]] (complete)
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|coveredblocks = M(4,2,1), [[M(4,2,3)]]<ref>For example consider block number 2 of <math>PSL_3(7) \triangleleft PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/].</ref> (complete)
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|pcoveringblocks = [[M(8,2,1)]], [[M(8,3,1)]], [[M(8,5,1)]] (complete)
 
}}
 
}}
  
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== Other notatable representatives ==
 
== Other notatable representatives ==
  
Block number 4 of <math>k PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]
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Block number 2 of <math>k PGL_3(7)</math> in the labelling used in [http://www.math.rwth-aachen.de/~MOC/decomposition/tex/L3(7)/]
 
 
== Covering blocks and covered blocks ==
 
 
 
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>.
 
 
 
If <math>b</math> lies in M(4,2,1), then <math>B</math> must lie in M(4,2,1) or [[M(4,2,3)]]. For example consider the principal blocks of <math>O_2(A_4) \triangleleft A_4</math>.
 
 
 
If <math>B</math> lies in M(4,2,1), then <math>b</math> must lie in M(4,2,1) or [[M(4,2,3)]]. For example consider blocks of <math>PSL_3(7) \triangleleft PGL_3(7)</math>.
 
  
 
== Projective indecomposable modules ==
 
== Projective indecomposable modules ==
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All irreducible characters have height zero.
 
All irreducible characters have height zero.
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[[C2xC2|Back to <math>C_2 \times C_2</math>]]
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== Notes ==
 +
 +
<references />

Latest revision as of 13:30, 19 December 2018

M(4,2,1) - [math]k(C_2 \times C_2)[/math]
M(4,2,1)quiver.png
Representative: [math]k(C_2 \times C_2)[/math]
Defect groups: [math]C_2 \times C_2[/math]
Inertial quotients: [math]1[/math]
[math]k(B)=[/math] 4
[math]l(B)=[/math] 1
[math]{\rm mf}_k(B)=[/math] 1
[math]{\rm Pic}_k(B)=[/math] [math](k \times k):GL_2(k)[/math]
Cartan matrix: [math]\left( \begin{array}{c} 4 \\ \end{array} \right)[/math]
Defect group Morita invariant? Yes
Inertial quotient Morita invariant? Yes
[math]\mathcal{O}[/math]-Morita classes known? Yes
[math]\mathcal{O}[/math]-Morita classes: [math]\mathcal{O} (C_2 \times C_2)[/math]
Decomposition matrices: [math]\left( \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]\mathcal{L}(B)=S_4[/math]
[math]PI(B)=[/math] [math]S_4 \times C_2[/math]
Source algebras known? Yes
Source algebra reps: [math]k(C_2 \times C_2)[/math]
[math]k[/math]-derived equiv. classes known? Yes
[math]k[/math]-derived equivalent to: Forms a derived equivalence class
[math]\mathcal{O}[/math]-derived equiv. classes known? Yes
[math]p'[/math]-index covering blocks: M(4,2,1), M(4,2,3) (complete)
[math]p'[/math]-index covered blocks: M(4,2,1), M(4,2,3)[1] (complete)
Index [math]p[/math] covering blocks: M(8,2,1), M(8,3,1), M(8,5,1) (complete)

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>, b:<1,1>

Relations w.r.t. [math]k[/math]: a^2=b^2=ab+ba=0

Other notatable representatives

Block number 2 of [math]k PGL_3(7)[/math] in the labelling used in [2]

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{ccc} & S_1 & \\ S_1 & & S_1 \\ & S_1 & \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_2 \times C_2[/math]

Notes

  1. For example consider block number 2 of [math]PSL_3(7) \triangleleft PGL_3(7)[/math] in the labelling used in [1].