M(32,51,19)

M(32,51,19) - [math]B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))[/math]

[[File: |250px]]

Representative:	[math]B_0(k({\rm Aut}(SL_2(8)) \times (C_2)^2))[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]C_7:C_3[/math]
[math]k(B)=[/math]	32
[math]l(B)=[/math]	5
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{ccccc} 16 & 8 & 8 & 16 & 8 \\ 8 & 16 & 8 & 16 & 8 \\ 8 & 8 & 16 & 16 & 8 \\ 16 & 16 & 16 & 32 & 12 \\ 8 & 8 & 8 & 12 & 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]B_0(\cO({\rm Aut}(SL_2(8)) \times (C_2)^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(32,51,17), M(32,51,18)
[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:

Other notatable representatives

[math]B_0(k((C_2)^2 \times {}^2G_2(q)))[/math], and the unique nonprincipal block with defect group $(C_2)^5$ of $(C_2)^2 \times \operatorname{Co}_3$.

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(32,51,19), then [math]B[/math] is in M(32,51,7), M(32,51,19), M(32,51,28) or M(32,51,34).

Projective indecomposable modules

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

[math]\begin{array}{ccc} \begin{array}{c} S_{1} \\ S_{1} S_{1} S_{4} \\ S_{3} S_{1} S_{1} S_{2} S_{4} S_{4} S_{5} \\ S_{3} S_{2} S_{1} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{2} S_{1} S_{2} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{3} S_{1} S_{1} S_{2} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{1} S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\ S_{1} S_{1} S_{4} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{2} S_{2} S_{4} \\ S_{2} S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\ S_{1} S_{3} S_{3} S_{2} S_{2} S_{1} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{2} S_{3} S_{1} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{1} S_{3} S_{3} S_{2} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{2} S_{3} S_{1} S_{4} S_{4} S_{5} \\ S_{2} S_{2} S_{4} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{3} S_{3} S_{4} \\ S_{1} S_{3} S_{3} S_{2} S_{4} S_{4} S_{5} \\ S_{1} S_{3} S_{2} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{1} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{1} S_{2} S_{2} S_{3} S_{3} S_{1} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{3} S_{4} S_{4} S_{5} \\ S_{3} S_{3} S_{4} \\ S_{3} \\ \end{array} \end{array}[/math]

 

[math] \begin{array}{cc} \begin{array}{c} S_{4} \\ S_{2} S_{1} S_{3} S_{4} S_{4} S_{5} \\ S_{1} S_{2} S_{1} S_{3} S_{2} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{1} S_{1} S_{2} S_{2} S_{3} S_{3} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{1} S_{2} S_{2} S_{3} S_{1} S_{1} S_{3} S_{1} S_{3} S_{2} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{1} S_{3} S_{2} S_{1} S_{1} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{1} S_{1} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{4} S_{4} S_{5} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{4} S_{5} S_{5} \\ S_{1} S_{2} S_{3} S_{4} S_{4} S_{5} \\ S_{1} S_{3} S_{3} S_{2} S_{2} S_{1} S_{4} S_{4} \\ S_{1} S_{2} S_{2} S_{3} S_{1} S_{3} S_{4} S_{4} \\ S_{3} S_{1} S_{2} S_{1} S_{3} S_{2} S_{4} S_{4} \\ S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\ S_{4} S_{5} S_{5} \\ S_{5} \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Decomposition matrix

[math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 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 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \end{array}\right)[/math]

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