Difference between revisions of "Blocks with basic algebras of low dimension"

From Block library
Jump to: navigation, search
([Completed classification with [LM20])
(Removed dim 9 non-example description)
 
(3 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
== Blocks with basic algebras of dimension at most 12 ==
+
== 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.
  
−
'''Quiver:''' a:<1,2>, b:<2,1>, c:<1,1>, d:<1,1>
+
The results are incorporated into the table below.   
−
 
 
−
'''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}
 
−
5 & 1 \\
 
−
1 & 2 \\
 
−
\end{array} \right)</math>
 
−
 
 
−
A corresponding <math>\mathcal{O}</math>-block would 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 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>
 
−
</ref>. This final case was ruled out in [[References#L|[LM20]]].
 
  
 
{| class="wikitable"
 
{| class="wikitable"
Line 99: Line 66:
 
|-
 
|-
 
| 12 || [[M(4,2,3)]] || <math>C_2 \times C_2</math> || <math>kA_4</math> || 4 || 3 ||
 
| 12 || [[M(4,2,3)]] || <math>C_2 \times C_2</math> || <math>kA_4</math> || 4 || 3 ||
 +
|-
 +
| 13 || [[M(16,7,3)]] || <math>D_{16}</math> || <math>B_0(kPGL_2(7))</math> || 7 || 2 ||
 +
|-
 +
| 13 || [[M(16,8,4)]] || <math>SD_{16}</math> || <math>B_3(k(3.M_{10}))</math> || 7 || 2 ||
 +
|-
 +
| 13 || [[M(7,1,7)]] || <math>C_7</math> || <math>B_{15}(k6.A_7)</math> || 5 || 3 ||
 +
|-
 +
| 13 || [[M(13,1,1)]] || <math>C_{13}</math> || <math>kC_{13}</math> || 13 || 1 ||
 +
|-
 +
| 13 || M(13,1,?) || <math>C_{13}</math> || <math>B_0(kPSL_3(3))</math> || 7 || 3 ||
 +
|-
 +
| 13 || M(17,1,?) || <math>C_{17}</math> || <math>B_0(kPSL_2(16))</math> || 10 || 2 ||
 +
|-
 +
| 14 || [[M(5,1,5)]] || <math>C_5</math> || <math>B_0(kS_5)</math> || 5 || 4 ||
 +
|-
 +
| 14 || [[M(7,1,2)]] || <math>C_7</math> || <math>kD_{14}</math> || 5 || 2 ||
 +
|-
 +
| 14 || [[M(7,1,5)]] || <math>C_7</math> || <math>B_0(kPSL_3(3))</math> || 5 || 3 ||
 +
|-
 +
| 14 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kPSL_2(37))</math> || 11 || 2 ||
 +
|-
 +
| 15 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kGL_3(7))</math> || ||
 +
|-
 +
| 15 || || <math>C_{13}</math> || ?? || || ||
 +
 
|}
 
|}
−
 
−
== Notes ==
 
−
 
−
<references />
 

Latest revision as of 13:33, 13 December 2023

Blocks with basic algebras of dimension at most 16

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 was identified. This final case was ruled out by Linckelmann and Murphy in [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 [Sa20]. Later Benson and Sambale in [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.

Dimension Class Defect group Representative [math]\dim_k(Z(A))[/math] [math]l(A)[/math] 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(9,2,23) [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
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
13 M(16,7,3) [math]D_{16}[/math] [math]B_0(kPGL_2(7))[/math] 7 2
13 M(16,8,4) [math]SD_{16}[/math] [math]B_3(k(3.M_{10}))[/math] 7 2
13 M(7,1,7) [math]C_7[/math] [math]B_{15}(k6.A_7)[/math] 5 3
13 M(13,1,1) [math]C_{13}[/math] [math]kC_{13}[/math] 13 1
13 M(13,1,?) [math]C_{13}[/math] [math]B_0(kPSL_3(3))[/math] 7 3
13 M(17,1,?) [math]C_{17}[/math] [math]B_0(kPSL_2(16))[/math] 10 2
14 M(5,1,5) [math]C_5[/math] [math]B_0(kS_5)[/math] 5 4
14 M(7,1,2) [math]C_7[/math] [math]kD_{14}[/math] 5 2
14 M(7,1,5) [math]C_7[/math] [math]B_0(kPSL_3(3))[/math] 5 3
14 M(19,1,?) [math]C_{19}[/math] [math]B_0(kPSL_2(37))[/math] 11 2
15 M(19,1,?) [math]C_{19}[/math] [math]B_0(kGL_3(7))[/math]
15 [math]C_{13}[/math]  ??
Retrieved from ‘http://wiki.manchester.ac.uk/blocks/index.php?title=Blocks_with_basic_algebras_of_low_dimension&oldid=1234’