Miscallaneous results
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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 occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9.
| 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(kPSL_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(kPSL_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 |
