M(32,51,12)

M(32,51,12) - [math]B_0(k(SL_2(16) \times C_2))[/math]

[[File: |250px]]

Representative:	[math]B_0(k(SL_2(16) \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]B_0(\mathcal{O}(SL_2(16) \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(32,51,11)
[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(32,51,12), then [math]B[/math] is also in M(32,51,12).

Irreducible characters

All irreducible characters have height zero.

Cartan matrix

[math]\left( \begin{array}{ccc} 32 & 16 & 16 & 16 & 16 & 8 & 8 & 8 & 8 & 8 & 8 & 4 & 4 & 4 & 4 \\ 16 & 16 & 8 & 8 & 8 & 4 & 8 & 4 & 4 & 8 & 0 & 0 & 2 & 4 & 0 \\ 16 & 8 & 16 & 8 & 8 & 8 & 4 & 0 & 4 & 4 & 8 & 0 & 0 & 2 & 4 \\ 16 & 8 & 8 & 16 & 8 & 4 & 8 & 8 & 0 & 4 & 4 & 4 & 0 & 0 & 2 \\ 16 & 8 & 8 & 8 & 16 & 8 & 4 & 4 & 8 & 0 & 4 & 2 & 4 & 0 & 0 \\ 8 & 4 & 8 & 4 & 8 & 8 & 2 & 0 & 4 & 0 & 4 & 0 & 0 & 0 & 0 \\ 8 & 8 & 4 & 8 & 4 & 2 & 8 & 4 & 0 & 4 & 0 & 0 & 0 & 0 & 0 \\ 8 & 4 & 0 & 8 & 4 & 0 & 4 & 8 & 0 & 2 & 0 & 4 & 0 & 0 & 0 \\ 8 & 4 & 4 & 0 & 8 & 4 & 0 & 0 & 8 & 0 & 2 & 0 & 4 & 0 & 0 \\ 8 & 8 & 4 & 4 & 0 & 0 & 4 & 2 & 0 & 8 & 0 & 0 & 0 & 4 & 0 \\ 8 & 0 & 8 & 4 & 4 & 4 & 0 & 0 & 2 & 0 & 8 & 0 & 0 & 0 & 4 \\ 4 & 0 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\ 4 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 \\ 4 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 \\ 4 & 0 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 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 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \end{array}\right)[/math]

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