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

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== Blocks with basic algebras of dimension at most 12 ==
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== Blocks with basic algebras of dimension at most 14 ==
  
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.  
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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 was identified<ref>The algebra of dimension 9 has the following structure.
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'''Quiver:''' a:<1,2>, b:<2,1>, c:<1,1>, d:<1,1>
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'''Relations w.r.t. <math>k</math>:''' ab=c^3=d^2, cd=dc=0, ca=bc=da=bd=0
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'''Cartan matrix:'''  <math>\left( \begin{array}{cc}
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5 & 1 \\
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1 & 2 \\
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\end{array} \right)</math>
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A corresponding <math>\mathcal{O}</math>-block would have '''decomposition matrix'''  <math>\left( \begin{array}{cc}
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1 & 0 \\
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1 & 0 \\
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1 & 0 \\
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1 & 0 \\
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0 & 1 \\
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1 & 1 \\
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\end{array}\right)</math>
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Labelling the simple modules by <math>S_1, S_2</math>, the projective indecomposable modules have Loewy structure as follows:
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<math>\begin{array}{cc}
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  \begin{array}{ccc}
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    & S_1 & \\
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      S_2 & \begin{array}{c} S_1 \\ S_1 \\ \end{array} & S_1 \\
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    & S_1 & \\
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\end{array} , &
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\begin{array}{c}
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      S_2 \\
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      S_1 \\
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      S_2 \\
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  \end{array}
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\end{array}
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</math>
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</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]]].
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The results are incorporated into the table below.
  
 
{| class="wikitable"
 
{| class="wikitable"
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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 ||
 
| 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 ||
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| 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
 
 
|-
 
|-
 
| 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||
 
| 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||
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|-
 
|-
 
| 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 ||
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|-
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| 13 || [[M(16,7,3)]] || <math>D_{16}</math> || <math>B_0(kPGL_2(7))</math> || 7 || 2 ||
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|-
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| 13 || [[M(16,8,4)]] || <math>SD_{16}</math> || <math>B_3(k(3.M_{10}))</math> || 7 || 2 ||
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|-
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| 13 || [[M(7,1,7)]] || <math>C_7</math> || <math>B_{15}(k6.A_7)</math> || 5 || 3 ||
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|-
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| 13 || [[M(13,1,1)]] || <math>C_{13}</math> || <math>kC_{13}</math> || 13 || 1 ||
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|-
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| 13 || M(13,1,?) || <math>C_{13}</math> || <math>B_0(kPSL_3(3))</math> || 7 || 3 ||
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|-
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| 13 || M(17,1,?) || <math>C_{17}</math> || <math>B_0(kPSL_2(16))</math> || 10 || 2 ||
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|-
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| 14 || [[M(5,1,5)]] || <math>C_5</math> || <math>B_0(kS_5)</math> || 5 || 4 ||
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|-
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| 14 || [[M(7,1,2)]] || <math>C_7</math> || <math>kD_{14}</math> || 5 || 2 ||
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|-
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| 14 || [[M(7,1,5)]] || <math>C_7</math> || <math>B_0(kPSL_3(3))</math> || 5 || 3 ||
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|-
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| 14 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kPSL_2(37))</math> || 11 || 2 ||
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|}
 
|}
  
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.
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== Notes ==
 
 
'''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.
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<references />

Revision as of 15:34, 11 August 2020

Blocks with basic algebras of dimension at most 14

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[1]. 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].

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

Notes

  1. The algebra of dimension 9 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=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]