Blocks with basic algebras of low dimension - Revision history

Charles Eaton: Removed dim 9 non-example description

2023-12-13T13:33:08Z

Removed dim 9 non-example description

Charles Eaton at 15:54, 10 October 2023

2023-10-10T15:54:37Z

Charles Eaton: /* Blocks with basic algebras of dimension at most 14 */

2023-10-10T15:46:09Z

‎Blocks with basic algebras of dimension at most 14

Charles Eaton: Dimensions 13 and 14

2020-08-11T15:34:42Z

Dimensions 13 and 14

Charles Eaton: [Completed classification with [LM20]

2020-05-23T08:29:20Z

[Completed classification with [LM20]

Charles Eaton: Corrected relations of algebra of dim 9

2019-04-10T08:47:37Z

Corrected relations of algebra of dim 9

Charles Eaton: /* Blocks with basic algebras of dimension at most 12 */ Renamed M(9,2,2) by M(9,2,23)

2018-10-14T11:12:09Z

‎Blocks with basic algebras of dimension at most 12: Renamed M(9,2,2) by M(9,2,23)

Charles Eaton: /* Blocks with basic algebras of dimension at most 12 */ SmallGroup(72,24), and changed to Q_8

2018-10-08T14:58:32Z

‎Blocks with basic algebras of dimension at most 12: SmallGroup(72,24), and changed to Q_8

Charles Eaton at 09:51, 23 September 2018

2018-09-23T09:51:36Z

Charles Eaton: Created page with "== Blocks with basic algebras of dimension at most 12 == In [Li18] Markus Linckelmann calculated the $k$ -algebras of dimension at most twelve which..."

2018-09-23T09:46:17Z

Created page with "== Blocks with basic algebras of dimension at most 12 == In [Li18] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which..."

New page

== Blocks with basic algebras of dimension at most 12 ==

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.

{| 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 ||
|}

The algebra <math>A</math> of dimension 9 which may or may not be the basic algebra for a <math>k</math>-block <math>B</math> of a finite group has the following structure.

'''Quiver:''' a:<1,2>, b:<2,1>, c:<1,1>, d:<1,1>

'''Relations w.r.t. <math>k</math>:''' ab=c^3=d^2, cd=dc=0, ca=bc=da=cd=0

'''Cartan matrix:''' <math>\left( \begin{array}{cc}
5 & 1 \\
1 & 2 \\
\end{array} \right)</math>

An <math>\mathcal{O}</math>-block corresponding to <math>B</math> must have '''decomposition matrix''' <math>\left( \begin{array}{cc}
1 & 0 \\
1 & 0 \\
1 & 0 \\
1 & 0 \\
0 & 1 \\
1 & 1 \\
\end{array}\right)</math>

Labelling the simple <math>B</math>-modules by <math>S_1, S_2</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{cc}
\begin{array}{ccc}
& S_1 & \\
S_2 & \begin{array}{c} S_1 \\ S_1 \\ \end{array} & S_1 \\
& S_1 & \\
\end{array} , &
\begin{array}{c}
S_2 \\
S_1 \\
S_2 \\
\end{array}
\end{array}
</math>

By [[References|[Ki84]]] the [[inertial quotient]] of <math>B</math> must either be <math>C_2</math> (acting with no non-trivial fixed points) or <math>D_8</math> (with a non-trivial 2-cocyle). By [[References|[Ko03]]] <math>B</math> cannot be Morita equivalent to a principal block.

@@ Line 1: / Line 1: @@
 == Blocks with basic algebras of dimension at most 16 ==
-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 was identified<ref>The algebra of dimension 9 has the following structure.
+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 was identified. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. Using the classification of finite simple groups, the basic algebras of dimension 13 or 14 for blocks of finite groups were calculated by Sambale in [[References#S|[Sa20]]]. Later Benson and Sambale in [[References#B|[BS23]]] gave a classification for dimensions 15 and 16, except for one unsettled case of a block with defect group <math>C_{13}</math> in dimension 15.
 The results are incorporated into the table below.
@@ Line 124: / Line 89: @@
 | 15 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kGL_3(7))</math> || ||
 |-
-| 15 || || <math>C_{13}</math> || ?? || ||
+| 15 || || <math>C_{13}</math> || ?? || || ||
 |}

@@ Line 1: / Line 1: @@
-== Blocks with basic algebras of dimension at most 14 ==
+== Blocks with basic algebras of dimension at most 16 ==
 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 was identified<ref>The algebra of dimension 9 has the following structure.
@@ Line 36: / Line 36: @@
 \end{array}
 </math>
-</ref>. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. Using the classification of finite simple groups, the basic algebras of dimension 13 or 14 for blocks of finite groups were calculated by Sambale in [[References#S|[Sa20]]].
+</ref>. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. Using the classification of finite simple groups, the basic algebras of dimension 13 or 14 for blocks of finite groups were calculated by Sambale in [[References#S|[Sa20]]]. Later Benson and Sambale in [[References#B|[BS23]]] gave a classification for dimensions 15 and 16, except for one unsettled case of a block with defect group <math>C_{13}</math> in dimension 15.
 The results are incorporated into the table below.

@@ Line 121: / Line 121: @@
 |-
 | 14 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kPSL_2(37))</math> || 11 || 2 ||
 |}

@@ Line 1: / Line 1: @@
-== Blocks with basic algebras of dimension at most 12 ==
+== Blocks with basic algebras of dimension at most 14 ==
 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 was identified<ref>The algebra of dimension 9 has the following structure.
@@ Line 36: / Line 36: @@
 \end{array}
 </math>
-</ref>. This final case was ruled out in [[References#L|[LM20]]].
+</ref>. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. Using the classification of finite simple groups, the basic algebras of dimension 13 or 14 for blocks of finite groups were calculated by Sambale in [[References#S|[Sa20]]].
 {| class="wikitable"
@@ Line 99: / Line 101: @@
 |-
 | 12 || [[M(4,2,3)]] || <math>C_2 \times C_2</math> || <math>kA_4</math> || 4 || 3 ||
 |}

@@ Line 1: / Line 1: @@
 == Blocks with basic algebras of dimension at most 12 ==
-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.
+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 was identified<ref>The algebra of dimension 9 has the following structure.
 {| class="wikitable"
@@ Line 50: / Line 85: @@
 |-
 | 9 || [[M(9,2,23)]] || <math>C_3 \times C_3</math> || Faithful block of <math>k((C_3 \times C_3):Q_8)</math>, in which <math>Z(Q_8)</math> acts trivially  || 6 || 1 || SmallGroup(72,24)
 |-
 | 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||
@@ Line 68: / Line 101: @@
 |}
-The algebra <math>A</math> of dimension 9 which may or may not be the basic algebra for a <math>k</math>-block <math>B</math> of a finite group has the following structure.
+== Notes ==
-By [[References|[Ki84]]] the [[inertial quotient]] of <math>B</math> must either be <math>C_2</math> (acting with no non-trivial fixed points) or <math>D_8</math> (with a non-trivial 2-cocyle). By [[References|[Ko03]]] <math>B</math> cannot be Morita equivalent to a principal block.
+<references />

@@ Line 72: / Line 72: @@
 '''Quiver:''' a:<1,2>, b:<2,1>, c:<1,1>, d:<1,1>
-'''Relations w.r.t. <math>k</math>:''' ab=c^3=d^2, cd=dc=0, ca=bc=da=cd=0
+'''Relations w.r.t. <math>k</math>:''' ab=c^3=d^2, cd=dc=0, ca=bc=da=bd=0
 '''Cartan matrix:'''   <math>\left( \begin{array}{cc}

@@ Line 49: / Line 49: @@
 | 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):Q_8)</math>, in which <math>Z(Q_8)</math> acts trivially  || 6 || 1 || SmallGroup(72,24)
+| 9 || [[M(9,2,23)]] || <math>C_3 \times C_3</math> || Faithful block of <math>k((C_3 \times C_3):Q_8)</math>, in which <math>Z(Q_8)</math> acts trivially  || 6 || 1 || SmallGroup(72,24)
 |-
 | 9 || || <math>C_3 \times C_3</math> || ? || 6 || 2 || Unknown