Difference between revisions of "Miscallaneous results"

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(Added M(9,2,2))
(→‎Morita invariance of the isomorphism type of a defect group: Section rewritten following counterexample to modular isomorphism problem)
 
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This page will contain results which do not fit in elsewhere on this site.
 
This page will contain results which do not fit in elsewhere on this site.
  
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== Blocks with basic algebras of dimension at most 12 ==
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== Blocks with basic algebras of low dimension ==
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[[Blocks with basic algebras of low dimension|Main article: Blocks with basic algebras of low dimension]]
  
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In [[References|[Li18]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9.
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In [[References#L|[Li18b]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. These results do not use the classification of finite simple groups. In [[References#S|[Sa20]]] Benjamin Sambale applied the classification of finite simple groups to extend the classification to dimensions 13 and 14. See [[Blocks with basic algebras of low dimension]] for a description of these results.
  
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{| class="wikitable"
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== Morita (non-)invariance of the isomorphism type of a defect group ==
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[[Morita invariance of the isomorphism type of a defect group|Main article: Morita invariance of the isomorphism type of a defect group]]
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! scope="col"| Dimension
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! scope="col"| Class
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In [[References#G|[GMdelR21]]] examples are given of non-isomorphic <math>2</math>-groups whose group algebras over a field of characteristic <math>2</math> are isomorphic, thus giving a counterexample to the modular isomorphism problem for fields of prime characteristic. This gives examples of blocks with non-isomorphic defect groups that are Morita equivalent. Note that these do no yield examples of Morita equivalent blocks defined over a local ring, so the question is still open as to whether the defect group is an invariant under Morita equivalence of such blocks. 
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! scope="col"| Defect group
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! scope="col"| Representative
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Note that the examples in [[References#G|[GMdelR21]]] also yield blocks that are Morita equivalent but not via a [[Glossary#Basic Morita/stable equivalence|basic Morita equivalence]].
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! scope="col"| <math>\dim_k(Z(A))</math>
 
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! scope="col"| <math>l(A)</math>
 
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! scope="col"| Notes
 
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| 1 || [[M(1,1,1)]] || <math>1</math> || <math>k1</math> || 1 || 1 || Blocks of defect zero
 
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|-
 
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| 2 || [[M(2,1,1)]] || <math>C_2</math> || <math>kC_2</math> || 2 || 1 ||
 
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|-
 
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| 3 || [[M(3,1,1)]] || <math>C_3</math> || <math>kC_3</math> || 3 || 1 ||
 
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|-
 
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| 4 || [[M(4,1,1)]] || <math>C_4</math> || <math>kC_4</math> || 4 || 1 ||
 
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|-
 
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| 4 || [[M(4,2,1)]] || <math>C_2 \times C_2</math> || <math>k(C_2 \times C_2)</math> || 4 || 1 ||
 
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|-
 
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| 5 || [[M(5,1,1)]] || <math>C_5</math> || <math>kC_5</math> || 5 || 1 ||
 
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| 6 || [[M(3,1,2)]] || <math>C_3</math> || <math>kS_3</math> || 3 || 2 ||
 
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| 7 || [[M(5,1,3)]] || <math>C_5</math> || <math>B_0(kA_5)</math> || 4 || 2 ||
 
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| 7 || [[M(7,1,1)]] || <math>C_7</math> || <math>kC_7</math> || 7 || 1 ||
 
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| 8 || [[M(8,1,1)]] || <math>C_8</math> || <math>kC_8</math> || 8 || 1 ||
 
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| 8 || [[M(8,2,1)]] || <math>C_4 \times C_2</math> || <math>k(C_4 \times C_2)</math> || 8 || 1 ||
 
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| 8 || [[M(8,3,1)]] || <math>D_8</math> || <math>kD_8</math> || 5 || 1 ||
 
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| 8 || [[M(8,4,1)]] || <math>Q_8</math> || <math>kQ_8</math> || 5 || 1 ||
 
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| 8 || [[M(8,5,1)]] || <math>C_2 \times C_2 \times C_2</math> || <math>k(C_2 \times C_2 \times C_2)</math> || 8 || 1 ||
 
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| 8 || [[M(7,1,3)]] || <math>C_7</math> || <math>B_0(kPSL_2(13))</math> || 5 || 2 ||
 
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| 9 || [[M(9,1,1)]] || <math>C_9</math> || <math>kC_9</math> || 9 || 1 ||
 
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| 9 || [[M(9,1,3)]] || <math>C_9</math> || <math>B_0(kPSL_2(8))</math> || 6 || 2 ||
 
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| 9 || [[M(9,2,1)]] || <math>C_3 \times C_3</math> || <math>k(C_3 \times C_3)</math> || 9 || 1 ||
 
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| 9 || [[M(9,2,2)]] || <math>C_3 \times C_3</math> || Faithful block of <math>k((C_3 \times C_3):D_8)</math>, in which <math>Z(D_8)</math> acts trivially  || 6 || 1 ||
 
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| 9 || || <math>C_3 \times C_3</math> || ? || 6 || 2 || Unknown
 
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| 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||
 
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|-
 
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| 10 || [[M(11,1,3)]] || <math>C_{11}</math> || <math>B_0(kPSL_2(32))</math> || 7 || 2 ||
 
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|-
 
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| 11 || [[M(8,3,3)]] || <math>D_8</math> || <math>kS_4</math> || 5 || 2 ||
 
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|-
 
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| 11 || [[M(7,1,6)]] || <math>C_7</math> || <math>B_0(kA_7)</math> || 5 || 3 ||
 
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| 11 || [[M(11,1,1)]] || <math>C_{11}</math> || <math>kC_{11}</math> || 11 || 1 ||
 
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|-
 
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| 11 || [[M(13,1,3)]] || <math>C_{13}</math> || <math>B_0(kPSL_2(25))</math> || 8 || 2 ||
 
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| 12 || [[M(4,2,3)]] || <math>C_2 \times C_2</math> || <math>kA_4</math> || 4 || 3 ||
 
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|}
 

Latest revision as of 13:51, 4 August 2022

This page will contain results which do not fit in elsewhere on this site.

Blocks with basic algebras of low dimension

Main article: Blocks with basic algebras of low dimension

In [Li18b] Markus Linckelmann calculated the [math]k[/math]-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified. This final case was ruled out by Linckelmann and Murphy in [LM20]. These results do not use the classification of finite simple groups. In [Sa20] Benjamin Sambale applied the classification of finite simple groups to extend the classification to dimensions 13 and 14. See Blocks with basic algebras of low dimension for a description of these results.

Morita (non-)invariance of the isomorphism type of a defect group

Main article: Morita invariance of the isomorphism type of a defect group

In [GMdelR21] examples are given of non-isomorphic [math]2[/math]-groups whose group algebras over a field of characteristic [math]2[/math] are isomorphic, thus giving a counterexample to the modular isomorphism problem for fields of prime characteristic. This gives examples of blocks with non-isomorphic defect groups that are Morita equivalent. Note that these do no yield examples of Morita equivalent blocks defined over a local ring, so the question is still open as to whether the defect group is an invariant under Morita equivalence of such blocks.

Note that the examples in [GMdelR21] also yield blocks that are Morita equivalent but not via a basic Morita equivalence.

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