M(32,51,11)

M(32,51,11) - [math]k(((C_2)^4 : C_{15}) \times C_2)[/math]

[[File: |250px]]

Representative:	[math]k(((C_2)^4 : C_{15}) \times C_2)[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]C_{15}[/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]\mathcal{O} (((C_2)^4 : C_{15}) \times C_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(32,51,12)
[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:

A block with defect group [math](C_2)^5[/math] and inertial quotient [math]C_{15}[/math] is Morita equivalent to M(32,51,11) or M(32,51,12), and the latter are derived equivalent.

It is unknown whether this Morita equivalence class contains blocks with inertial quotient [math]C_7:C_3 \times C_3[/math] (with action as in M(32,51,24)).

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(32,51,11), then [math]B[/math] is in M(32,51,2), M(32,51,5), or M(32,51,11).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_{15}[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccc} \begin{array}{c} S_1 \\ S_1 S_{10} S_{11} S_9 S_4 \\ S_{11} S_9 S_4 S_{10} S_{12} S_3 S_{14} S_{15} S_2 S_{13} \\ S_3 S_{12} S_{14} S_2 S_{15} S_{13} S_5 S_6 S_8 S_7 \\ S_6 S_5 S_7 S_8 S_1 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_{12} S_{14} S_5 S_7 S_2 \\ S_{10} S_7 S_{14} S_6 S_8 S_4 S_1 S_5 S_{12} S_{15} \\ S_{15} S_6 S_1 S_8 S_4 S_{10} S_3 S_9 S_{13} S_{11} \\ S_9 S_{13} S_3 S_{11} S_2 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_3 S_7 S_2 S_6 S_{11} \\ S_{12} S_7 S_{11} S_5 S_2 S_4 S_6 S_1 S_{14} S_{13} \\ S_{14} S_4 S_1 S_5 S_{12} S_{13} S_{15} S_9 S_8 S_{10} \\ S_9 S_8 S_{10} S_{15} S_3 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_{13} S_9 S_{14} S_{15} S_4 \\ S_{14} S_2 S_8 S_6 S_3 S_{10} S_{15} S_5 S_9 S_{13} \\ S_8 S_{10} S_1 S_5 S_2 S_7 S_3 S_6 S_{12} S_{11} \\ S_7 S_1 S_{12} S_{11} S_4 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_1 S_{10} S_{12} S_5 S_8 \\ S_4 S_7 S_1 S_{10} S_8 S_9 S_3 S_{15} S_{12} S_{11} \\ S_{14} S_{11} S_{13} S_2 S_4 S_7 S_{15} S_3 S_9 S_6 \\ S_{13} S_{14} S_6 S_5 S_2 \\ S_5 \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_6 \\ S_1 S_5 S_{11} S_6 S_{13} \\ S_{12} S_{11} S_{10} S_1 S_5 S_2 S_8 S_9 S_4 S_{13} \\ S_9 S_3 S_{10} S_{14} S_{15} S_8 S_7 S_2 S_4 S_{12} \\ S_7 S_{14} S_{15} S_3 S_6 \\ S_6 \\ \end{array} & \begin{array}{c} S_7 \\ S_4 S_6 S_{14} S_1 S_7 \\ S_5 S_9 S_{13} S_{14} S_4 S_6 S_{11} S_{15} S_1 S_{10} \\ S_3 S_{13} S_8 S_2 S_9 S_{12} S_{15} S_5 S_{11} S_{10} \\ S_{12} S_2 S_7 S_3 S_8 \\ S_7 \\ \end{array} & \begin{array}{c} S_8 \\ S_8 S_1 S_3 S_7 S_9 \\ S_9 S_{14} S_1 S_4 S_{10} S_6 S_2 S_{11} S_7 S_3 \\ S_{14} S_{15} S_6 S_2 S_4 S_{11} S_{10} S_{13} S_{12} S_5 \\ S_{15} S_5 S_{13} S_8 S_{12} \\ S_8 \\ \end{array} & \begin{array}{c} S_9 \\ S_3 S_{14} S_{10} S_2 S_9 \\ S_{10} S_{15} S_2 S_7 S_6 S_{14} S_{12} S_{11} S_5 S_3 \\ S_4 S_6 S_5 S_{12} S_{13} S_{11} S_7 S_1 S_{15} S_8 \\ S_1 S_{13} S_8 S_4 S_9 \\ S_9 \\ \end{array} & \begin{array}{c} S_{10} \\ S_{15} S_{12} S_3 S_{10} S_{11} \\ S_4 S_2 S_{13} S_8 S_7 S_6 S_3 S_{15} S_{11} S_{12} \\ S_9 S_5 S_7 S_{14} S_6 S_{13} S_2 S_8 S_4 S_1 \\ S_{14} S_5 S_9 S_1 S_{10} \\ S_{10} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{11} \\ S_{13} S_{11} S_2 S_{12} S_4 \\ S_8 S_4 S_9 S_7 S_{12} S_{15} S_2 S_{13} S_{14} S_5 \\ S_1 S_{14} S_8 S_3 S_5 S_9 S_6 S_{15} S_{10} S_7 \\ S_1 S_{10} S_6 S_3 S_{11} \\ S_{11} \\ \end{array} & \begin{array}{c} S_{12} \\ S_{15} S_4 S_{12} S_7 S_8 \\ S_6 S_3 S_4 S_9 S_1 S_7 S_{14} S_{13} S_{15} S_8 \\ S_2 S_{13} S_3 S_1 S_9 S_6 S_{10} S_{11} S_{14} S_5 \\ S_{11} S_5 S_2 S_{10} S_{12} \\ S_{12} \\ \end{array} & \begin{array}{c} S_{13} \\ S_8 S_5 S_{13} S_9 S_2 \\ S_3 S_{12} S_7 S_1 S_5 S_{10} S_9 S_8 S_2 S_{14} \\ S_{12} S_7 S_3 S_{10} S_{14} S_1 S_{11} S_{15} S_6 S_4 \\ S_{11} S_4 S_{15} S_6 S_{13} \\ S_{13} \\ \end{array} & \begin{array}{c} S_{14} \\ S_6 S_{10} S_{14} S_{15} S_5 \\ S_{15} S_5 S_3 S_{12} S_{13} S_{11} S_1 S_{10} S_8 S_6 \\ S_2 S_{12} S_8 S_{13} S_3 S_1 S_{11} S_4 S_9 S_7 \\ S_7 S_4 S_9 S_2 S_{14} \\ S_{14} \\ \end{array} & \begin{array}{c} S_{15} \\ S_{15} S_6 S_8 S_{13} S_3 \\ S_2 S_8 S_{13} S_3 S_5 S_1 S_6 S_9 S_{11} S_7 \\ S_2 S_{11} S_7 S_5 S_1 S_9 S_{10} S_4 S_{12} S_{14} \\ S_{14} S_{12} S_{10} S_4 S_{15} \\ S_{15} \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Cartan matrix

[math]\left( \begin{array}{ccc} 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 \end{array} \right)[/math]

Decomposition matrix

[math]\left( \begin{array}{ccc} 1 & 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 & 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 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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 & 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 & 0 & 0 & 0 & 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 & 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 & 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 & 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 & 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 & 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 & 0 & 0 & 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 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 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 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 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 & 1 & 1 \end{array}\right)[/math]

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