M(32,51,29)
| Representative: | [math]B_0(k({\rm Aut}(SL_2(8)) \times A_5))[/math] |
|---|---|
| Defect groups: | [math](C_2)^5[/math] |
| Inertial quotients: | [math](C_{7}:C_3) \times C_3[/math] |
| [math]k(B)=[/math] | 32 |
| [math]l(B)=[/math] | 15 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | |
| Cartan matrix: | See below. |
| Defect group Morita invariant? | Yes |
| Inertial quotient Morita invariant? | Yes |
| [math]\mathcal{O}[/math]-Morita classes known? | Yes |
| [math]\mathcal{O}[/math]-Morita classes: | [math]B_0(\mathcal{O}({\rm Aut}(SL_2(8)) \times A_5))[/math] |
| Decomposition matrices: | See below. |
| [math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
| [math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | |
| [math]PI(B)=[/math] | |
| Source algebras known? | No |
| Source algebra reps: | |
| [math]k[/math]-derived equiv. classes known? | Yes |
| [math]k[/math]-derived equivalent to: | M(32,51,24), M(32,51,25), M(32,51,26), M(32,51,27), M(32,51,28) |
| [math]\mathcal{O}[/math]-derived equiv. classes known? | Yes |
| [math]p'[/math]-index covering blocks: | |
| [math]p'[/math]-index covered blocks: | |
| Index [math]p[/math] covering blocks: |
Contents
Basic algebra
Other notatable representatives
Covering blocks and covered blocks
Let [math]N \triangleleft G[/math] with prime [math]p'[/math]-index and let [math]B[/math] be a block of [math]\mathcal{O} G[/math] covering a block [math]b[/math] of [math]\mathcal{O} N[/math].
If [math]b[/math] is in M(32,51,29), then [math]B[/math] is in M(32,51,16) or M(32,51,29).
Projective indecomposable modules
Irreducible characters
All irreducible characters have height zero.
Cartan matrix
[math]\left( \begin{array}{ccccccccccccccccccccc} 16 & 8 & 8 & 4 & 4 & 8 & 8 & 4 & 4 & 16 & 8 & 8 & 8 & 4 & 4 \\ 8 & 16 & 8 & 8 & 4 & 4 & 4 & 4 & 8 & 16 & 8 & 8 & 8 & 4 & 4 \\ 8 & 8 & 16 & 4 & 8 & 4 & 4 & 8 & 4 & 16 & 8 & 8 & 8 & 4 & 4 \\ 4 & 8 & 4 & 8 & 2 & 4 & 2 & 4 & 4 & 8 & 4 & 8 & 4 & 4 & 2 \\ 4 & 4 & 8 & 2 & 8 & 2 & 4 & 4 & 4 & 8 & 4 & 4 & 8 & 2 & 4 \\ 8 & 4 & 4 & 4 & 2 & 8 & 4 & 4 & 2 & 8 & 4 & 8 & 4 & 4 & 2 \\ 8 & 4 & 4 & 2 & 4 & 4 & 8 & 2 & 4 & 8 & 4 & 4 & 8 & 2 & 4 \\ 4 & 4 & 8 & 4 & 4 & 4 & 2 & 8 & 2 & 8 & 4 & 8 & 4 & 4 & 2 \\ 4 & 8 & 4 & 4 & 4 & 2 & 4 & 2 & 8 & 8 & 4 & 4 & 8 & 2 & 4 \\ 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 32 & 12 & 16 & 16 & 6 & 6 \\ 8 & 8 & 8 & 4 & 4 & 4 & 4 & 4 & 4 & 12 & 8 & 6 & 6 & 4 & 4 \\ 8 & 8 & 8 & 8 & 4 & 8 & 4 & 8 & 4 & 16 & 6 & 16 & 8 & 6 & 3 \\ 8 & 8 & 8 & 4 & 8 & 4 & 8 & 4 & 8 & 16 & 6 & 8 & 16 & 3 & 6 \\ 4 & 4 & 4 & 4 & 2 & 4 & 2 & 4 & 2 & 6 & 4 & 6 & 3 & 4 & 2 \\ 4 & 4 & 4 & 2 & 4 & 2 & 4 & 2 & 4 & 6 & 4 & 3 & 6 & 2 & 4 \end{array}\right)[/math]
Decomposition matrix
[math]\left( \begin{array}{ccccccccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 2 & 1 & 2 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 2 & 1 & 0 & 2 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 1 & 1 \end{array}\right)[/math]
