M(32,51,5)

M(32,51,5) - [math]k((C_2)^4 : C_5) \times C_2)[/math]

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Representative:	[math]k((C_2)^4 : C_5) \times C_2)[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]C_5[/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}{ccc} 8 & 6 & 6 & 6 & 6 \\ 6 & 8 & 6 & 6 & 6 \\ 6 & 6 & 8 & 6 & 6 \\ 6 & 6 & 6 & 8 & 6 \\ 6 & 6 & 6 & 6 & 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]\mathcal{O} ((C_2)^4 : C_5) \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:	Forms a derived equivalence class
[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_5[/math] is Morita equivalent to M(32,51,5).

It is unknown whether this Morita equivalence class contains blocks with inertial quotient [math]C_7:C_3[/math] (with action as in M(32,51,20)).

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(32,51,5), then [math]B[/math] is in M(32,51,1), M(32,51,5), or M(32,51,11).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3, S_4, S_5[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccc} \begin{array}{c} S_1 \\ S_1 S_5 S_2 S_4 S_3 \\ S_3 S_5 S_4 S_2 S_1 S_4 S_3 S_1 S_5 S_2 \\ S_2 S_2 S_3 S_5 S_4 S_1 S_3 S_5 S_1 S_4 \\ S_4 S_5 S_2 S_3 S_1 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_4 S_2 S_1 S_5 S_3 \\ S_5 S_1 S_3 S_1 S_4 S_4 S_2 S_2 S_5 S_3 \\ S_2 S_3 S_1 S_5 S_4 S_2 S_3 S_4 S_5 S_1 \\ S_3 S_4 S_1 S_5 S_2 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_5 S_4 S_2 S_3 S_1 \\ S_4 S_2 S_3 S_3 S_4 S_1 S_2 S_1 S_5 S_5 \\ S_4 S_2 S_3 S_2 S_1 S_1 S_4 S_5 S_5 S_3 \\ S_5 S_2 S_4 S_1 S_3 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_5 S_1 S_2 S_3 S_4 \\ S_1 S_3 S_2 S_2 S_3 S_5 S_5 S_1 S_4 S_4 \\ S_5 S_3 S_4 S_4 S_3 S_2 S_5 S_2 S_1 S_1 \\ S_3 S_5 S_2 S_1 S_4 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_2 S_4 S_5 S_1 S_3 \\ S_5 S_3 S_5 S_2 S_4 S_2 S_3 S_4 S_1 S_1 \\ S_2 S_4 S_1 S_5 S_2 S_5 S_1 S_4 S_3 S_3 \\ S_2 S_3 S_1 S_4 S_5 \\ S_5 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Decomposition matrix

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

Back to [math](C_2)^5[/math]