Difference between revisions of "Miscallaneous results"
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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. | 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. | ||
| + | |||
| + | {| 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 || | ||
| + | |} | ||
Revision as of 08:39, 22 September 2018
This page will contain results which do not fit in elsewhere on this site.
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 |
