M(32,51,1)

M(32,51,1) - [math]k((C_2)^5)[/math]

[[File: |250px]]

Representative:	[math]k((C_2)^5)[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]1[/math]
[math]k(B)=[/math]	32
[math]l(B)=[/math]	1
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{c} 32 \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)[/math]
Decomposition matrices:	[math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math]	1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math]	[math](C_2)^5:GL_5(2)[/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:

These are nilpotent blocks.

Covering blocks and covered blocks

Let [math]N \triangleleft G[/math] with [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(32,51,1), then [math]B[/math] is in M(32,51,1), M(32,51,2), M(32,51,4), M(32,51,6), M(32,51,8), M(16,14,11), M(16,14,13), M(16,14,17), M(16,14,20),M(16,14,22),M(16,14,24) or M(16,14,30).

Projective indecomposable modules

Labelling the unique simple [math]B[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{c} S_1 \\ S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 \\ S_1 \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

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