Difference between revisions of "M(32,51,19)"
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|inertial-morita-inv? = Yes | |inertial-morita-inv? = Yes | ||
|O-morita? = Yes | |O-morita? = Yes | ||
| − | |O-morita = <math>B_0(\ | + | |O-morita = <math>B_0(\mathcal{O}({\rm Aut}(SL_2(8)) \times (C_2)^2))</math> |
|decomp = See below. | |decomp = See below. | ||
|O-morita-frob = 1 | |O-morita-frob = 1 | ||
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== Other notatable representatives == | == Other notatable representatives == | ||
| − | <math>B_0(k((C_2)^2 \times {}^2G_2(q)))</math>, and the unique nonprincipal block with defect group | + | <math>B_0(k((C_2)^2 \times {}^2G_2(q)))</math>, and the unique nonprincipal block with defect group <math>(C_2)^5</math> of <math>(C_2)^2 \times \operatorname{Co}_3</math>. |
| − | |||
== Covering blocks and covered blocks == | == Covering blocks and covered blocks == | ||
Latest revision as of 18:49, 8 December 2019
| Representative: | [math]B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))[/math] |
|---|---|
| Defect groups: | [math](C_2)^5[/math] |
| Inertial quotients: | [math]C_7:C_3[/math] |
| [math]k(B)=[/math] | 32 |
| [math]l(B)=[/math] | 5 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | |
| Cartan matrix: | [math]\left( \begin{array}{ccccc} 16 & 8 & 8 & 16 & 8 \\ 8 & 16 & 8 & 16 & 8 \\ 8 & 8 & 16 & 16 & 8 \\ 16 & 16 & 16 & 32 & 12 \\ 8 & 8 & 8 & 12 & 8 \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]B_0(\mathcal{O}({\rm Aut}(SL_2(8)) \times (C_2)^2))[/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(32,51,17), M(32,51,18) |
| [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
[math]B_0(k((C_2)^2 \times {}^2G_2(q)))[/math], and the unique nonprincipal block with defect group [math](C_2)^5[/math] of [math](C_2)^2 \times \operatorname{Co}_3[/math].
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(32,51,19), then [math]B[/math] is in M(32,51,7), M(32,51,19), M(32,51,28) or M(32,51,34).
Projective indecomposable modules
Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_5[/math], the projective indecomposable modules have Loewy structure as follows:
[math]\begin{array}{ccc} \begin{array}{c} S_{1} \\ S_{1} S_{1} S_{4} \\ S_{3} S_{1} S_{1} S_{2} S_{4} S_{4} S_{5} \\ S_{3} S_{2} S_{1} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{2} S_{1} S_{2} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{3} S_{1} S_{1} S_{2} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{1} S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\ S_{1} S_{1} S_{4} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{2} S_{2} S_{4} \\ S_{2} S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\ S_{1} S_{3} S_{3} S_{2} S_{2} S_{1} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{2} S_{3} S_{1} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{1} S_{3} S_{3} S_{2} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{2} S_{3} S_{1} S_{4} S_{4} S_{5} \\ S_{2} S_{2} S_{4} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{3} S_{3} S_{4} \\ S_{1} S_{3} S_{3} S_{2} S_{4} S_{4} S_{5} \\ S_{1} S_{3} S_{2} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{1} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{1} S_{2} S_{2} S_{3} S_{3} S_{1} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{3} S_{4} S_{4} S_{5} \\ S_{3} S_{3} S_{4} \\ S_{3} \\ \end{array} \end{array}[/math]
[math] \begin{array}{cc} \begin{array}{c} S_{4} \\ S_{2} S_{1} S_{3} S_{4} S_{4} S_{5} \\ S_{1} S_{2} S_{1} S_{3} S_{2} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{1} S_{1} S_{2} S_{2} S_{3} S_{3} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{1} S_{2} S_{2} S_{3} S_{1} S_{1} S_{3} S_{1} S_{3} S_{2} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{1} S_{3} S_{2} S_{1} S_{1} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{1} S_{1} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{4} S_{4} S_{5} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{4} S_{5} S_{5} \\ S_{1} S_{2} S_{3} S_{4} S_{4} S_{5} \\ S_{1} S_{3} S_{3} S_{2} S_{2} S_{1} S_{4} S_{4} \\ S_{1} S_{2} S_{2} S_{3} S_{1} S_{3} S_{4} S_{4} \\ S_{3} S_{1} S_{2} S_{1} S_{3} S_{2} S_{4} S_{4} \\ S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\ S_{4} S_{5} S_{5} \\ S_{5} \\ \end{array} \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.
Decomposition matrix
[math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 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 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \end{array}\right)[/math]
