Difference between revisions of "Blocks with basic algebras of low dimension"
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== Blocks with basic algebras of dimension at most 12 == | == 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 | + | 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. |
| + | |||
| + | '''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> | ||
| + | </ref>. This final case was ruled out in [[References#L|[LM20]]]. | ||
{| class="wikitable" | {| class="wikitable" | ||
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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 || [[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) | ||
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| 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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| − | + | == Notes == | |
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Revision as of 09:29, 23 May 2020
Blocks with basic algebras of dimension at most 12
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 in [LM20].
| 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 |
Notes
- ↑ 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]
