M(16,14,7)

M(16,14,7) - [math]B_0(k(\C_2 \times SL_2(8)))[/math]

[[File: |250px]]

Representative:	[math]B_0(k(\C_2 \times SL_2(8)))[/math]
Defect groups:	[math](C_2)^4[/math]
Inertial quotients:	[math]C_7[/math]
[math]k(B)=[/math]	8
[math]l(B)=[/math]	7
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{ccc} 16 & 8 & 8 & 8 & 4 & 4 & 4 \\ & 8 & 8 & 4 & 4 & 4 & 0 & 2 \\ & 8 & 4 & 8 & 4 & 2 & 4 & 0 \\ & 8 & 4 & 4 & 8 & 0 & 2 & 4 \\ & 4 & 4 & 2 & 0 & 4 & 0 & 0 \\ & 4 & 0 & 4 & 2 & 0 & 4 & 0 \\ & 4 & 2 & 0 & 4 & 0 & 0 & 4 \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_0(\mathcal{O}(\C_2 \times SL_2(8)))[/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(16,14,6)
[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:

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(16,14,7), then [math]B[/math] is in M(16,14,7) or M(16,14,15).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3, S_4, S_5, S_6, S_7[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccccc} \begin{array}{c} S_1 \\ S_1 S_2 S_3 S_4\\ S_1 S_1 S_1 S_2 S_3 S_4 S_5 S_6 S_7 \\ S_1 S_1 S_1 S_2 S_2 S_3 S_3 S_4 S_4 S_5 S_6 S_7 S_1 S_1 S_1 S_2 S_2 S_3 S_3 S_4 S_4 S_5 S_6 S_7\\ S_1 S_1 S_1 S_1 S_2 S_3 S_4 S_5 S_6 S_7 \\ S_1 S_2 S_3 S_4\\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_1 S_2 S_5 \\ S_1 S_2 S_3 S_4 S_5 \\ S_1 S_1 S_2 S_3 S_4 S_7 \\ S_1 S_1 S_2 S_3 S_4 S_7 \\ S_1 S_2 S_3 S_4 S_5 \\ S_1 S_2 S_5 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_1 S_3 S_6 \\ S_1 S_2 S_3 S_4 S_6 \\ S_1 S_1 S_2 S_3 S_4 S_5 \\ S_1 S_1 S_2 S_3 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_6 \\ S_1 S_3 S_6 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_1 S_4 S_7 \\ S_1 S_2 S_3 S_4 S_7 \\ S_1 S_1 S_2 S_3 S_4 S_6 \\ S_1 S_1 S_2 S_3 S_4 S_6 \\ S_1 S_2 S_3 S_4 S_7 \\ S_1 S_4 S_7 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_2 S_5 \\ S_1 S_2 \\ S_1 S_3 \\ S_1 S_3 \\ S_1 S_2 \\ S_2 S_5 S_5 \\ \end{array} & \begin{array}{c} S_6 \\ S_3 S_6 \\ S_1 S_3 \\ S_1 S_4 \\ S_1 S_4 \\ S_1 S_3 \\ S_3 S_6 S_6 \\ \end{array} & \begin{array}{c} S_7 \\ S_4 S_7 \\ S_1 S_4 \\ S_1 S_2 \\ S_1 S_2 \\ S_1 S_4 \\ S_4 S_7 S_7 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Decomposition matrix

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

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