M(32,51,31)

M(32,51,31) - [math]B_0(k({\rm Aut}SL_2(32)))[/math]

[[File: |250px]]

Representative:	[math]B_0(k({\rm Aut}SL_2(32)))[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]C_{31}:C_5[/math]
[math]k(B)=[/math]	16
[math]l(B)=[/math]	11
[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}({\rm Aut}SL_2(32)))[/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(32,51,30)
[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 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(32,51,31), then [math]B[/math] is in M(32,51,23) or M(32,51,31).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_{11}[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccc} \begin{array}{c} S_{1} \\ S_{6} \\ S_{5} S_{4} S_{3} S_{2} S_{1} S_{7} S_{8} \\ S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\ S_{2} S_{2} S_{1} S_{4} S_{3} S_{5} S_{1} S_{3} S_{5} S_{4} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{9} S_{10} \\ S_{5} S_{5} S_{1} S_{3} S_{4} S_{3} S_{1} S_{2} S_{4} S_{2} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\ S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\ S_{6} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{6} \\ S_{4} S_{3} S_{2} S_{5} S_{1} S_{7} S_{8} \\ S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\ S_{4} S_{5} S_{3} S_{2} S_{2} S_{3} S_{4} S_{1} S_{1} S_{5} S_{7} S_{8} S_{8} S_{7} S_{7} S_{8} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\ S_{3} S_{5} S_{2} S_{5} S_{4} S_{4} S_{1} S_{1} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\ S_{4} S_{5} S_{2} S_{3} S_{1} S_{7} S_{8} \\ S_{6} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{6} \\ S_{5} S_{4} S_{3} S_{1} S_{2} S_{8} S_{7} \\ S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\ S_{5} S_{1} S_{5} S_{2} S_{1} S_{4} S_{4} S_{2} S_{3} S_{3} S_{8} S_{8} S_{7} S_{7} S_{7} S_{8} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{9} S_{10} \\ S_{4} S_{3} S_{3} S_{5} S_{5} S_{2} S_{1} S_{2} S_{4} S_{1} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\ S_{4} S_{1} S_{3} S_{5} S_{2} S_{7} S_{8} \\ S_{6} \\ S_{3} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{4} \\ S_{6} \\ S_{3} S_{2} S_{5} S_{4} S_{1} S_{8} S_{7} \\ S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\ S_{1} S_{5} S_{2} S_{2} S_{3} S_{3} S_{1} S_{5} S_{4} S_{4} S_{7} S_{8} S_{8} S_{8} S_{7} S_{7} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{9} \\ S_{5} S_{3} S_{1} S_{4} S_{4} S_{2} S_{2} S_{1} S_{5} S_{3} S_{8} S_{7} S_{7} S_{8} S_{7} S_{8} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\ S_{5} S_{3} S_{4} S_{1} S_{2} S_{8} S_{7} \\ S_{6} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{6} \\ S_{1} S_{5} S_{4} S_{3} S_{2} S_{8} S_{7} \\ S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\ S_{5} S_{3} S_{4} S_{4} S_{1} S_{5} S_{2} S_{1} S_{2} S_{3} S_{8} S_{7} S_{8} S_{8} S_{7} S_{7} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} S_{9} \\ S_{4} S_{2} S_{2} S_{3} S_{5} S_{5} S_{1} S_{3} S_{1} S_{4} S_{7} S_{7} S_{8} S_{8} S_{8} S_{7} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} \\ S_{2} S_{5} S_{3} S_{4} S_{1} S_{8} S_{7} \\ S_{6} \\ S_{5} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{6} \\ S_{4} S_{1} S_{3} S_{2} S_{5} S_{7} S_{8} S_{8} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\ S_{5} S_{1} S_{1} S_{3} S_{3} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{2} S_{5} S_{4} S_{1} S_{5} S_{2} S_{4} S_{4} S_{4} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} S_{10} S_{9} S_{9} S_{10} \\ S_{5} S_{3} S_{1} S_{2} S_{3} S_{4} S_{5} S_{5} S_{2} S_{5} S_{2} S_{5} S_{2} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{3} S_{4} S_{5} S_{4} S_{1} S_{2} S_{1} S_{4} S_{3} S_{1} S_{1} S_{7} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{8} S_{7} S_{7} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{7} S_{7} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} S_{10} S_{10} S_{10} S_{9} S_{9} S_{10} \\ S_{2} S_{4} S_{5} S_{4} S_{5} S_{4} S_{3} S_{4} S_{3} S_{2} S_{3} S_{1} S_{3} S_{5} S_{2} S_{1} S_{1} S_{5} S_{1} S_{2} S_{7} S_{7} S_{8} S_{7} S_{8} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\ S_{2} S_{1} S_{5} S_{4} S_{3} S_{8} S_{8} S_{7} \\ S_{6} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{7} \\ S_{6} S_{9} S_{10} \\ S_{5} S_{1} S_{4} S_{3} S_{2} S_{8} S_{7} S_{7} S_{8} S_{11} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} S_{10} \\ S_{4} S_{4} S_{5} S_{1} S_{5} S_{3} S_{4} S_{2} S_{5} S_{1} S_{2} S_{3} S_{3} S_{2} S_{1} S_{7} S_{7} S_{8} S_{7} S_{8} S_{8} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{9} \\ S_{1} S_{2} S_{1} S_{1} S_{4} S_{4} S_{2} S_{2} S_{5} S_{3} S_{4} S_{3} S_{3} S_{5} S_{5} S_{8} S_{7} S_{8} S_{7} S_{8} S_{7} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{9} \\ S_{1} S_{3} S_{5} S_{4} S_{2} S_{7} S_{7} S_{8} S_{8} S_{11} \\ S_{6} S_{10} S_{9} \\ S_{7} \\ \end{array} & \begin{array}{c} S_{8} \\ S_{6} S_{6} S_{10} \\ S_{1} S_{4} S_{5} S_{2} S_{3} S_{7} S_{8} S_{7} S_{8} S_{8} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\ S_{4} S_{5} S_{2} S_{4} S_{5} S_{5} S_{4} S_{2} S_{1} S_{1} S_{3} S_{1} S_{3} S_{3} S_{2} S_{8} S_{8} S_{7} S_{8} S_{8} S_{8} S_{7} S_{8} S_{7} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{10} S_{10} \\ S_{4} S_{4} S_{4} S_{2} S_{1} S_{1} S_{5} S_{1} S_{3} S_{3} S_{2} S_{2} S_{5} S_{5} S_{3} S_{8} S_{8} S_{7} S_{7} S_{8} S_{8} S_{7} S_{8} S_{8} \\ S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{6} S_{10} S_{9} \\ S_{3} S_{5} S_{4} S_{1} S_{2} S_{8} S_{7} S_{7} S_{8} S_{8} \\ S_{6} S_{6} S_{10} \\ S_{8} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{9} \\ S_{7} S_{11} \\ S_{6} S_{10} S_{9} \\ S_{3} S_{1} S_{4} S_{5} S_{2} S_{8} S_{7} \\ S_{6} S_{6} S_{6} \\ S_{2} S_{5} S_{3} S_{5} S_{1} S_{4} S_{3} S_{4} S_{1} S_{2} S_{7} \\ S_{6} S_{6} S_{6} \\ S_{4} S_{3} S_{2} S_{1} S_{5} S_{8} S_{7} \\ S_{6} S_{9} S_{10} \\ S_{7} S_{11} \\ S_{9} \\ \end{array} & \begin{array}{c} S_{10} \\ S_{7} S_{8} \\ S_{6} S_{6} S_{9} \\ S_{2} S_{4} S_{3} S_{1} S_{5} S_{8} S_{7} S_{7} \\ S_{6} S_{6} S_{6} S_{6} S_{10} \\ S_{1} S_{3} S_{3} S_{5} S_{5} S_{2} S_{4} S_{4} S_{1} S_{2} S_{8} S_{8} S_{8} \\ S_{6} S_{6} S_{6} S_{6} S_{10} \\ S_{3} S_{2} S_{1} S_{5} S_{4} S_{7} S_{7} S_{8} \\ S_{6} S_{6} S_{9} \\ S_{8} S_{7} \\ S_{10} \\ \end{array} & \begin{array}{c} S_{11} \\ S_{9} \\ S_{7} \\ S_{6} \\ S_{1} S_{4} S_{3} S_{5} S_{2} \\ S_{6} \\ S_{2} S_{1} S_{3} S_{4} S_{5} \\ S_{6} \\ S_{7} \\ S_{9} \\ S_{11} \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Cartan matrix

[math]\left( \begin{array}{ccc} 8 & 6 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\ 6 & 8 & 6 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\ 6 & 6 & 8 & 6 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\ 6 & 6 & 6 & 8 & 6 & 16 & 8 & 8 & 4 & 4 & 2 \\ 6 & 6 & 6 & 6 & 8 & 16 & 8 & 8 & 4 & 4 & 2 \\ 16 & 16 & 16 & 16 & 16 & 48 & 20 & 28 & 8 & 12 & 3 \\ 8 & 8 & 8 & 8 & 8 & 20 & 12 & 10 & 5 & 6 & 2 \\ 8 & 8 & 8 & 8 & 8 & 28 & 10 & 20 & 2 & 7 & 0 \\ 4 & 4 & 4 & 4 & 4 & 8 & 5 & 2 & 4 & 2 & 2 \\ 4 & 4 & 4 & 4 & 4 & 12 & 6 & 7 & 2 & 4 & 0 \\ 2 & 2 & 2 & 2 & 2 & 3 & 2 & 0 & 2 & 0 & 2 \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 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 4 & 1 & 3 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 2 & 1 & 0 & 1 & 0 & 1 \end{array}\right)[/math]

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