Difference between revisions of "M(32,51,33)"
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{{blockbox | {{blockbox | ||
| − | |title = M(32,51, | + | |title = M(32,51,33) - <math>b_2(k((C_2)^5:(C_7:3^{1+2}_+)))</math> |
|image = | |image = | ||
|representative = <math>b_2(k((C_2)^5:(C_7:3^{1+2}_+)))</math> | |representative = <math>b_2(k((C_2)^5:(C_7:3^{1+2}_+)))</math> | ||
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|source? = No | |source? = No | ||
|sourcereps = | |sourcereps = | ||
| − | |k-derived-known? = | + | |k-derived-known? = No |
|k-derived = | |k-derived = | ||
| − | |O-derived-known? = | + | |O-derived-known? = No |
|coveringblocks = | |coveringblocks = | ||
|coveredblocks = | |coveredblocks = | ||
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This Morita equivalence class contains only non-principal blocks. | This Morita equivalence class contains only non-principal blocks. | ||
| + | |||
| + | It is unknown whether this class is derived equivalent to [[M(32,51,34)]]; if not, it forms its own derived equivalence class. | ||
== Basic algebra == | == Basic algebra == | ||
Latest revision as of 15:08, 9 December 2019
| Representative: | [math]b_2(k((C_2)^5:(C_7:3^{1+2}_+)))[/math] |
|---|---|
| Defect groups: | [math](C_2)^5[/math] |
| Inertial quotients: | [math]C_7:C_3 \times C_3[/math] |
| [math]k(B)=[/math] | 16 |
| [math]l(B)=[/math] | 7 |
| [math]{\rm mf}_k(B)=[/math] | 1 |
| [math]{\rm Pic}_k(B)=[/math] | |
| Cartan matrix: | [math]\left( \begin{array}{ccccccc} 8 & 4 & 4 & 4 & 4 & 4 & 4 \\ 4 & 6 & 5 & 5 & 4 & 4 & 4 \\ 4 & 5 & 6 & 5 & 4 & 4 & 4 \\ 4 & 5 & 5 & 6 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 & 6 & 5 & 5 \\ 4 & 4 & 4 & 4 & 5 & 6 & 5 \\ 4 & 4 & 4 & 4 & 5 & 5 & 6 \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_2(k((C_2)^5:(C_7:3^{1+2}_+)))[/math] |
| Decomposition matrices: | [math]\left( \begin{array}{ccccccc} 1 & 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 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 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 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 0 & 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 & 1 & 1 & 1 \end{array}\right)[/math] |
| [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? | No |
| [math]k[/math]-derived equivalent to: | |
| [math]\mathcal{O}[/math]-derived equiv. classes known? | No |
| [math]p'[/math]-index covering blocks: | |
| [math]p'[/math]-index covered blocks: | |
| Index [math]p[/math] covering blocks: |
This Morita equivalence class contains only non-principal blocks.
It is unknown whether this class is derived equivalent to M(32,51,34); if not, it forms its own derived equivalence class.
Contents
Basic algebra
Other notatable representatives
Any nonprincipal block of [math](C_2)^5:(C_7:3^{1+2}_-)[/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(32,51,33), then [math]B[/math] is in M(32,51,13), M(32,51,17) or M(32,51,33).
Projective indecomposable modules
Labelling the simple [math]b_2[/math]-modules by [math]S_1,\dots, S_7 [/math], the projective indecomposable modules have Loewy structure as follows:
[math]\begin{array}{cccc} \begin{array}{c} S_{1} \\ S_{1} S_{4} S_{1} S_{6} S_{7} \\ S_{2} S_{7} S_{4} S_{3} S_{4} S_{6} S_{6} S_{5} S_{7} S_{1} \\ S_{5} S_{4} S_{2} S_{6} S_{3} S_{2} S_{3} S_{1} S_{7} S_{5} \\ S_{1} S_{2} S_{5} S_{3} S_{1} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{1} S_{3} S_{6} S_{5} S_{3} \\ S_{7} S_{1} S_{2} S_{2} S_{1} S_{7} S_{5} S_{5} S_{4} S_{4} \\ S_{2} S_{6} S_{6} S_{7} S_{1} S_{3} S_{6} S_{3} S_{2} S_{4} \\ S_{7} S_{3} S_{5} S_{5} S_{4} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{5} S_{2} S_{1} S_{5} S_{7} \\ S_{3} S_{4} S_{2} S_{3} S_{1} S_{2} S_{1} S_{4} S_{6} S_{6} \\ S_{3} S_{5} S_{5} S_{1} S_{3} S_{6} S_{4} S_{7} S_{7} S_{7} \\ S_{6} S_{4} S_{2} S_{2} S_{5} \\ S_{3} \\ \end{array} \end{array}[/math]
[math] \begin{array}{ccc} \begin{array}{c} S_{4} \\ S_{2} S_{7} S_{7} S_{3} S_{6} \\ S_{1} S_{5} S_{4} S_{2} S_{4} S_{3} S_{5} S_{5} S_{6} S_{6} \\ S_{4} S_{3} S_{7} S_{1} S_{7} S_{4} S_{1} S_{3} S_{2} S_{2} \\ S_{6} S_{7} S_{1} S_{6} S_{5} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{4} S_{3} S_{2} S_{1} S_{2} \\ S_{5} S_{3} S_{6} S_{5} S_{6} S_{1} S_{1} S_{3} S_{7} S_{7} \\ S_{4} S_{6} S_{2} S_{5} S_{2} S_{7} S_{5} S_{1} S_{4} S_{4} \\ S_{6} S_{3} S_{7} S_{3} S_{2} \\ S_{5} \\ \end{array} & \begin{array}{c} S_{6} \\ S_{3} S_{5} S_{7} S_{4} S_{4} \\ S_{1} S_{6} S_{2} S_{2} S_{3} S_{2} S_{5} S_{6} S_{7} S_{7} \\ S_{3} S_{6} S_{1} S_{5} S_{6} S_{1} S_{5} S_{3} S_{4} S_{4} \\ S_{1} S_{7} S_{2} S_{7} S_{4} \\ S_{6} \\ \end{array} & \begin{array}{c} S_{7} \\ S_{2} S_{4} S_{6} S_{5} S_{6} \\ S_{4} S_{3} S_{7} S_{7} S_{1} S_{5} S_{2} S_{4} S_{3} S_{3} \\ S_{1} S_{7} S_{6} S_{2} S_{1} S_{6} S_{2} S_{7} S_{5} S_{5} \\ S_{4} S_{4} S_{1} S_{6} S_{3} \\ S_{7} \\ \end{array} \end{array} [/math]
Irreducible characters
All irreducible characters have height zero.
