M(16,14,8)
| Representative: | [math]k(A_4 \times A_4)[/math] |
|---|---|
| Defect groups: | [math](C_2)^4[/math] |
| Inertial quotients: | [math]C_3 \times C_3[/math] |
| [math]k(B)=[/math] | 16 |
| [math]l(B)=[/math] | 9 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | |
| Cartan matrix: | [math]\left( \begin{array}{ccccccccc} 4 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 2 \\ 2 & 4 & 1 & 2 & 1 & 2 & 1 & 1 & 2 \\ 2 & 1 & 4 & 2 & 2 & 1 & 1 & 2 & 1 \\ 1 & 2 & 2 & 4 & 2 & 2 & 1 & 1 & 1 \\ 1 & 1 & 2 & 2 & 4 & 1 & 2 & 1 & 2 \\ 1 & 2 & 1 & 2 & 1 & 4 & 2 & 2 & 1 \\ 1 & 1 & 1 & 1 & 2 & 2 & 4 & 2 & 2 \\ 2 & 1 & 2 & 1 & 1 & 2 & 2 & 4 & 1 \\ 2 & 2 & 1 & 1 & 2 & 1 & 2 & 1 & 4 \end{array} \right)[/math] |
| 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]\mathcal{O} (A_4 \times A_4)[/math] |
| Decomposition matrices: | See below |
| [math]{\rm mf}_\mathcal{O}(B)=[/math] | 1 |
| [math]{\rm Pic}_{\mathcal{O}}(B)=[/math] | [math]S_3 \wr C_2[/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(16,14,9), M(16,14,10) |
| [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(16,14,8), then [math]B[/math] is in M(16,14,3), M(16,14,4), M(16,14,8).
Projective indecomposable modules
Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_9[/math], the projective indecomposable modules have Loewy structure as follows:
[math]\begin{array}{ccccccccc} \begin{array}{c} S_1 \\ S_2 S_3 S_8 S_9 \\ S_1 S_1 S_4 S_5 S_6 S_7 \\ S_2 S_3 S_8 S_9 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_1 S_4 S_6 S_9 \\ S_2 S_2 S_3 S_5 S_7 S_8 \\ S_1 S_4 S_6 S_9 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_1 S_4 S_5 S_8 \\ S_2 S_3 S_3 S_6 S_7 S_9 \\ S_1 S_4 S_5 S_8 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_2 S_3 S_5 S_6 \\ S_1 S_4 S_4 S_7 S_8 S_9 \\ S_2 S_3 S_5 S_6 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_3 S_4 S_7 S_9 \\ S_1 S_2 S_5 S_5 S_6 S_8 \\ S_3 S_4 S_7 S_9 \\ S_5 \\ \end{array} & \begin{array}{c} S_7 \\ S_5 S_6 S_8 S_9 \\ S_1 S_2 S_3 S_4 S_7 S_7 \\ S_5 S_6 S_8 S_9 \\ S_7 \\ \end{array} & \begin{array}{c} S_8 \\ S_1 S_3 S_6 S_7 \\ S_2 S_4 S_5 S_8 S_8 S_9 \\ S_1 S_3 S_6 S_7 \\ S_8 \\ \end{array} & \begin{array}{c} S_9 \\ S_1 S_2 S_5 S_7 \\ S_3 S_4 S_6 S_8 S_9 S_9 \\ S_1 S_2 S_5 S_7 \\ S_9 \\ \end{array} \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.
Decomposition matrix
[math]\left( \begin{array}{ccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 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 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 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 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right)[/math]
