Difference between revisions of "M(32,51,20)"
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| − | The action of the inertial quotient on the defect group is such that the subgroup | + | The action of the inertial quotient on the defect group is such that the subgroup <math>C_3</math> acts as in [[M(32,51,5)]]. |
== Basic algebra == | == Basic algebra == | ||
Latest revision as of 11:42, 9 December 2019
| Representative: | [math]k((C_2)^5 : (C_7:C_3))[/math] |
|---|---|
| Defect groups: | [math](C_2)^5[/math] |
| Inertial quotients: | [math]C_7:C_3[/math] |
| [math]k(B)=[/math] | 16 |
| [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} 4 & 2 & 2 & 4 & 4 \\ 2 & 4 & 2 & 4 & 4 \\ 2 & 2 & 4 & 4 & 4 \\ 4 & 4 & 4 & 16 & 12 \\ 4 & 4 & 4 & 12 & 16 \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}((C_2)^5 : (C_7:C_3))[/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(32,51,21) |
| [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: |
The action of the inertial quotient on the defect group is such that the subgroup [math]C_3[/math] acts as in M(32,51,5).
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(32,51,20), then [math]B[/math] is in M(32,51,6), M(32,51,17), M(32,51,24).
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_{3} S_{2} S_{4} \\ S_{1} S_{4} S_{4} S_{5} \\ S_{1} S_{5} S_{5} S_{4} \\ S_{2} S_{3} S_{5} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{1} S_{3} S_{4} \\ S_{2} S_{4} S_{4} S_{5} \\ S_{2} S_{5} S_{5} S_{4} \\ S_{3} S_{1} S_{5} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{2} S_{1} S_{4} \\ S_{3} S_{4} S_{4} S_{5} \\ S_{3} S_{5} S_{5} S_{4} \\ S_{2} S_{1} S_{5} \\ S_{3} \\ \end{array} \end{array}[/math]
[math] \begin{array}{cc} \begin{array}{c} S_{4} \\ S_{5} S_{4} S_{5} S_{4} S_{4} \\ S_{3} S_{2} S_{1} S_{5} S_{5} S_{4} S_{5} S_{4} S_{5} S_{5} S_{4} S_{4} \\ S_{1} S_{2} S_{3} S_{2} S_{3} S_{1} S_{4} S_{4} S_{5} S_{4} S_{5} S_{5} S_{5} S_{4} \\ S_{2} S_{1} S_{3} S_{4} S_{5} S_{4} S_{4} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{1} S_{2} S_{3} S_{4} S_{5} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{1} S_{2} S_{3} S_{5} S_{4} S_{4} S_{4} S_{5} S_{4} S_{5} S_{5} \\ S_{1} S_{3} S_{2} S_{5} S_{4} S_{5} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{5} S_{4} 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 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ 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 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 3 & 3 \end{array}\right)[/math]
