Block library - User contributions [en-gb]

Generic classifications by p-group class

2019-12-10T14:36:31Z

Elliotmckernon: /* Homocyclic 2-groups when inertial quotient contains a Singer cycle */

This page will contain results for generic classes of ''p''-groups. It is very much under construction so the list below is not complete.

== Cyclic ''p''-groups ==

[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].

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.

For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.

[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]

[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]

== Tame blocks ==

[[Tame blocks|Click here for background on tame blocks]].

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.

== Abelian ''2''-groups with ''2''-rank at most three ==

[[Image:under-construction.png|50px|left]]

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.

Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math>

{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
| <strong><math>rk_2(D) \leq 3</math>  </strong>
|-
! scope="col"| <math>D</math>
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes
! scope="col"| Always endopermutation source Morita equivalent?
! scope="col"| References
! scope="col"| Notes
|-
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]
|-
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||
|-
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||
|-
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]
|-
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||
|-
|[[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]]] ||
|-
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||
|-
|[[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]]] ||
|-
|[[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]]
|}

== Abelian ''2''-groups ==

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.

== Minimal nonabelian <math>2</math>-groups ==

[[Image:under-construction.png|50px|left]]

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>.

For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.

== Homocyclic ''2''-groups when inertial quotient contains a Singer cycle ==

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>.

<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>.

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.

Generic classifications by p-group class

2019-12-10T14:02:13Z

Elliotmckernon: Adding section on homoyclic defect groups when the inertial quotient contains a Singer cycle

This page will contain results for generic classes of ''p''-groups. It is very much under construction so the list below is not complete.

== Cyclic ''p''-groups ==

[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].

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.

For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.

[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]

[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]

== Tame blocks ==

[[Tame blocks|Click here for background on tame blocks]].

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.

== Abelian ''2''-groups with ''2''-rank at most three ==

[[Image:under-construction.png|50px|left]]

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.

Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math>

{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
| <strong><math>rk_2(D) \leq 3</math>  </strong>
|-
! scope="col"| <math>D</math>
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes
! scope="col"| Always endopermutation source Morita equivalent?
! scope="col"| References
! scope="col"| Notes
|-
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]
|-
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||
|-
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||
|-
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]
|-
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||
|-
|[[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]]] ||
|-
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||
|-
|[[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]]] ||
|-
|[[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]]
|}

== Abelian ''2''-groups ==

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.

== Minimal nonabelian <math>2</math>-groups ==

[[Image:under-construction.png|50px|left]]

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>.

For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.

== Homocyclic ''2''-groups when inertial quotient contains a Singer cycle ==

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>.

<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, the inertial quotient satisfies <math>C_{2^n-1} \triangleleft \mathbb{E} \leq C_{2^n-1} : 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 <math> \mathcal{O}J_1 </math>, which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}SL_2(2^n) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. With the exception of <math>\mathcal{O}J_1</math>, the Morita equivalence between the block and the class representative is known to be basic.

(C2)^4

2019-08-22T16:16:04Z

Elliotmckernon:

__NOTITLE__

== Blocks with defect group <math>(C_2)^4</math> ==

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]]].

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Representative
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]
! scope="col"| <math>k(B)</math>
! scope="col"| <math>l(B)</math>
! scope="col"| Inertial quotients
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math>
! scope="col"| <math>{\rm Pic}_k(B)</math>
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math>
! scope="col"| <math>{\rm mf_k(B)}</math>
! scope="col"| Notes

|-
|[[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 ||
|-
|[[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 ||
|-
|[[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 ||
|-
|[[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>
|-
|[[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>
|-
|[[M(16,14,6)]] || <math>k(C_2 \times ((C_2)^3:C_7))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 ||
|-
|[[M(16,14,7)]] || <math>B_0(k(C_2 \times SL_2(8)))</math> || 1 ||16 ||7 ||<math>C_7</math> || || ||1 ||1 ||
|-
|[[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 ||
|-
|[[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 ||
|-
|[[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 ||
|-
|[[M(16,14,11)]] || <math>k((C_2)^4 : C_{15})</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}</math> || ||1 ||1 ||
|-
|[[M(16,14,12)]] || <math>B_0(kSL_2(16))</math> || 1 ||16 ||15 ||<math>C_{15}</math> || <math>C_{15}:C_4</math> || ||1 ||1 ||
|-
|[[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 ||
|-
|[[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 ||
|-
|[[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 ||
|-
|[[M(16,14,16)]] || <math>b(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.
|}

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.

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>.
In particular:

[[M(16,14,2)]] and [[M(16,14,3)]] are derived equivalent over <math>\mathcal{O}</math>.

[[M(16,14,6)]] and [[M(16,14,7)]] are derived equivalent over <math>\mathcal{O}</math>.

[[M(16,14,8)]] [[M(16,14,9)]] and [[M(16,14,10)]] are derived equivalent over <math>\mathcal{O}</math>.

[[M(16,14,11)]] and [[M(16,14,12)]] are derived equivalent over <math>\mathcal{O}</math>.

[[M(16,14,13)]], [[M(16,14,14)]] and [[M(16,14,15)]] are derived equivalent over <math>\mathcal{O}</math>.