Miscallaneous results - Revision history

Charles Eaton: /* Morita invariance of the isomorphism type of a defect group */ Section rewritten following counterexample to modular isomorphism problem

2022-08-04T13:51:27Z

‎Morita invariance of the isomorphism type of a defect group: Section rewritten following counterexample to modular isomorphism problem

Charles Eaton: Added [Sa20] results

2020-08-11T14:36:26Z

Added [Sa20] results

Charles Eaton at 08:17, 23 May 2020

2020-05-23T08:17:57Z

Charles Eaton: References to work in [LM20]

2020-05-23T08:17:21Z

References to work in [LM20]

Charles Eaton: Defect groups Morita invariant

2019-01-03T15:58:14Z

Defect groups Morita invariant

Charles Eaton at 10:02, 24 September 2018

2018-09-24T10:02:03Z

Charles Eaton: Moved Linckelmann result to its own page

2018-09-23T09:20:49Z

Moved Linckelmann result to its own page

Charles Eaton at 20:49, 22 September 2018

2018-09-22T20:49:53Z

Charles Eaton: Added M(9,2,2)

2018-09-22T20:46:50Z

Added M(9,2,2)

Charles Eaton at 17:56, 22 September 2018

2018-09-22T17:56:40Z

@@ Line 6: / Line 6: @@
 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.
-== Morita invariance of the isomorphism type of a defect group ==
+== Morita (non-)invariance of the isomorphism type of a defect group ==
 [[Morita invariance of the isomorphism type of a defect group|Main article: Morita invariance of the isomorphism type of a defect group]]
-It is not known whether there exist Morita equivalent blocks with non-isomorphic defect groups. In general this is a difficult problem, subsuming the modular isomorphism problem for <math>p</math>-groups. [[Glossary#Basic Morita/stable equivalence|Basic Morita equivalences]] do preserve the isomorphism type of a defect group, and part of the difficulty in resolving the question is the lack of examples of Morita equivalent blocks which are not also basic Morita equivalent (this is not to say that every known Morita equivalence is basic). A survey may be found in [[References#N|[NS18]]].
+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.

@@ Line 1: / Line 1: @@
 This page will contain results which do not fit in elsewhere on this site.
-== Blocks with basic algebras of dimension at most 12 ==
+== Blocks with basic algebras of low dimension ==
 [[Blocks with basic algebras of low dimension|Main article: Blocks with basic algebras of low dimension]]
-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 in [[References#L|[LM20]]]. See [[Blocks with basic algebras of low dimension]] for a description of these results.
+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.
 == Morita invariance of the isomorphism type of a defect group ==

@@ Line 4: / Line 4: @@
 [[Blocks with basic algebras of low dimension|Main article: Blocks with basic algebras of low dimension]]
-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 in [[References#L|[LM20]]]]See [[Blocks with basic algebras of low dimension]] for a description of these results.
+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 in [[References#L|[LM20]]]. See [[Blocks with basic algebras of low dimension]] for a description of these results.
 == Morita invariance of the isomorphism type of a defect group ==

@@ Line 4: / Line 4: @@
 [[Blocks with basic algebras of low dimension|Main article: Blocks with basic algebras of low dimension]]
-In [[References|[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. See [[Blocks with basic algebras of low dimension]] for a description of these results.
+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 in [[References#L|[LM20]]]]See [[Blocks with basic algebras of low dimension]] for a description of these results.
 == Morita invariance of the isomorphism type of a defect group ==

← Older revision		Revision as of 15:58, 3 January 2019
Line 5:		Line 5:

	In [[References\|[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. See [[Blocks with basic algebras of low dimension]] for a description of these results.		In [[References\|[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. See [[Blocks with basic algebras of low dimension]] for a description of these results.
		+
		+	== Morita invariance of the isomorphism type of a defect group ==
		+	[[Morita invariance of the isomorphism type of a defect group\|Main article: Morita invariance of the isomorphism type of a defect group]]
		+
		+	It is not known whether there exist Morita equivalent blocks with non-isomorphic defect groups. In general this is a difficult problem, subsuming the modular isomorphism problem for <math>p</math>-groups. [[Glossary#Basic Morita/stable equivalence\|Basic Morita equivalences]] do preserve the isomorphism type of a defect group, and part of the difficulty in resolving the question is the lack of examples of Morita equivalent blocks which are not also basic Morita equivalent (this is not to say that every known Morita equivalence is basic). A survey may be found in [[References#N\|[NS18]]].

← Older revision		Revision as of 10:02, 24 September 2018
Line 4:		Line 4:
	[[Blocks with basic algebras of low dimension\|Main article: Blocks with basic algebras of low dimension]]		[[Blocks with basic algebras of low dimension\|Main article: Blocks with basic algebras of low dimension]]

−	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 where no block with that basic algebra is identified. See [[Blocks with basic algebras of low dimension]] for a description of these results.	+	In [[References\|[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. See [[Blocks with basic algebras of low dimension]] for a description of these results.

← Older revision		Revision as of 09:20, 23 September 2018
Line 2:		Line 2:

	== Blocks with basic algebras of dimension at most 12 ==		== Blocks with basic algebras of dimension at most 12 ==
		+	[[Blocks with basic algebras of low dimension\|Main article: Blocks with basic algebras of low dimension]]

−	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.	+	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 where no block with that basic algebra is identified. See [[Blocks with basic algebras of low dimension]] for a description of these results.
−
−	~~{\| class="wikitable"~~
−	\|-
−	~~! scope="col"\| Dimension~~
−	~~! scope="col"\| Class~~
−	~~! scope="col"\| Defect group~~
−	~~! scope="col"\| Representative~~
−	~~! scope="col"\| <math>\dim_k(Z(A))</math>~~
−	~~! scope="col"\| <math>l(A)</math>~~
−	~~! scope="col"\| Notes~~
−	\|-
−	~~\| 1 \|\|~~ [[~~M(1,1,1)]] \|\| <math>1</math> \|\| <math>k1</math> \|\| 1 \|\| 1 \|\|~~ Blocks of ~~defect zero~~
−	\|-
−	~~\| 2 \|\| [[M(2,1,1)]] \|\| <math>C_2</math> \|\| <math>kC_2</math> \|\| 2 \|\| 1 \|\|~~
−	\|-
−	~~\| 3 \|\| [[M(3,1,1)]] \|\| <math>C_3</math> \|\| <math>kC_3</math> \|\| 3 \|\| 1 \|\|~~
−	\|-
−	~~\| 4 \|\| [[M(4,1,1)]] \|\| <math>C_4</math> \|\| <math>kC_4</math> \|\| 4 \|\| 1 \|\|~~
−	\|-
−	~~\| 4 \|\| [[M(4,2,1)]] \|\| <math>C_2 \times C_2</math> \|\| <math>k(C_2 \times C_2)</math> \|\| 4 \|\| 1 \|\|~~
−	\|-
−	~~\| 5 \|\| [[M(5,1,1)]] \|\| <math>C_5</math> \|\| <math>kC_5</math> \|\| 5 \|\| 1 \|\|~~
−	\|-
−	~~\| 6 \|\| [[M(3,1,2)]] \|\| <math>C_3</math> \|\| <math>kS_3</math> \|\| 3 \|\| 2 \|\|~~
−	\|-
−	~~\| 7 \|\| [[M(5,1,3)]] \|\| <math>C_5</math> \|\| <math>B_0(kA_5)</math> \|\| 4 \|\| 2 \|\|~~
−	\|-
−	~~\| 7 \|\| [[M(7,1,1)]] \|\| <math>C_7</math> \|\| <math>kC_7</math> \|\| 7 \|\| 1 \|\|~~
−	\|-
−	~~\| 8 \|\| [[M(8,1,1)]] \|\| <math>C_8</math> \|\| <math>kC_8</math> \|\| 8 \|\| 1 \|\|~~
−	\|-
−	~~\| 8 \|\| [[M(8,2,1)~~]] ~~\|\| <math>C_4 \times C_2</math> \|\| <math>k(C_4 \times C_2)</math> \|\| 8 \|\| 1 \|\|~~
−	\|-
−	~~\| 8 \|\| [[M(8,3,1)]] \|\| <math>D_8</math> \|\| <math>kD_8</math> \|\| 5 \|\| 1 \|\|~~
−	\|-
−	~~\| 8 \|\| [[M(8,4,1)]] \|\| <math>Q_8</math> \|\| <math>kQ_8</math> \|\| 5 \|\| 1 \|\|~~
−	\|-
−	~~\| 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 \|\|~~
−	\|-
−	~~\| 8 \|\| [[M(7,1,3)]] \|\| <math>C_7</math> \|\| <math>B_0(kPSL_2(13))</math> \|\| 5 \|\| 2 \|\|~~
−	\|-
−	~~\| 9 \|\| [[M(9,1,1)]] \|\| <math>C_9</math> \|\| <math>kC_9</math> \|\| 9 \|\| 1 \|\|~~
−	\|-
−	~~\| 9 \|\| [[M(9,1,3)]] \|\| <math>C_9</math> \|\| <math>B_0(kSL_2(8))</math> \|\| 6 \|\| 2 \|\|~~
−	\|-
−	~~\| 9 \|\| [[M(9,2,1)]] \|\| <math>C_3 \times C_3</math> \|\| <math>k(C_3 \times C_3)</math> \|\| 9 \|\| 1 \|\|~~
−	\|-
−	~~\| 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 \|\|~~
−	\|-
−	~~\| 9 \|\| \|\| <math>C_3 \times C_3</math> \|\| ? \|\| 6 \|\| 2 \|\| Unknown~~
−	\|-
−	~~\| 10 \|\| [[M(5,1,2)]] \|\| <math>C_5</math> \|\| <math>kD_{10}</math> \|\| 4 \|\| 2 \|\|~~
−	\|-
−	~~\| 10 \|\| [[M(11,1,3)]] \|\| <math>C_{11}</math> \|\| <math>B_0(kSL_2(32))</math> \|\| 7 \|\| 2 \|\|~~
−	\|-
−	~~\| 11 \|\| [[M(8,3,3)]] \|\| <math>D_8</math> \|\| <math>kS_4</math> \|\| 5 \|\| 2 \|\|~~
−	\|-
−	~~\| 11 \|\| [[M(7,1,6)]] \|\| <math>C_7</math> \|\| <math>B_0(kA_7)</math> \|\| 5 \|\| 3 \|\|~~
−	\|-
−	~~\| 11 \|\| [[M(11,1,1)]] \|\| <math>C_{11}</math> \|\| <math>kC_{11}</math> \|\| 11 \|\| 1 \|\|~~
−	\|-
−	~~\| 11 \|\| [[M(13,1,3)]] \|\| <math>C_{13}</math> \|\| <math>B_0(kPSL_2(25))</math> \|\| 8 \|\| 2 \|\|~~
−	\|-
−	~~\| 12 \|\| [[M(4,2,3)]] \|\| <math>C_2 \times C_2</math> \|\| <math>kA_4</math> \|\| 4 \|\| 3 \|\|~~
−	\|}

@@ Line 47: / Line 47: @@
 | 9 || [[M(9,1,1)]] || <math>C_9</math> || <math>kC_9</math> || 9 || 1 ||
 |-
-| 9 || [[M(9,1,3)]] || <math>C_9</math> || <math>B_0(kPSL_2(8))</math> || 6 || 2 ||
+| 9 || [[M(9,1,3)]] || <math>C_9</math> || <math>B_0(kSL_2(8))</math> || 6 || 2 ||
 |-
 | 9 || [[M(9,2,1)]] || <math>C_3 \times C_3</math> || <math>k(C_3 \times C_3)</math> || 9 || 1 ||
@@ Line 57: / Line 57: @@
 | 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||
 |-
-| 10 || [[M(11,1,3)]] || <math>C_{11}</math> || <math>B_0(kPSL_2(32))</math> || 7 || 2 ||
+| 10 || [[M(11,1,3)]] || <math>C_{11}</math> || <math>B_0(kSL_2(32))</math> || 7 || 2 ||
 |-
 | 11 || [[M(8,3,3)]] || <math>D_8</math> || <math>kS_4</math> || 5 || 2 ||

@@ Line 50: / Line 50: @@
 |-
 | 9 || [[M(9,2,1)]] || <math>C_3 \times C_3</math> || <math>k(C_3 \times C_3)</math> || 9 || 1 ||
 |-
 | 9 || || <math>C_3 \times C_3</math> || ? || 6 || 2 || Unknown

@@ Line 32: / Line 32: @@
 |-
 | 7 || [[M(7,1,1)]] || <math>C_7</math> || <math>kC_7</math> || 7 || 1 ||
 |}