# M(32,51,31)

M(32,51,31) - $B_0(k({\rm Aut}SL_2(32)))$
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Representative: $B_0(k({\rm Aut}SL_2(32)))$ $(C_2)^5$ $C_{31}:C_5$ 16 11 1 See below. Yes Yes Yes $B_0(\mathcal{O}({\rm Aut}SL_2(32)))$ See below. 1 No Yes M(32,51,30) Yes

## Covering blocks and covered blocks

Let $N \triangleleft G$ with prime $p'$-index and let $B$ be a block of $\mathcal{O} G$ covering a block $b$ of $\mathcal{O} N$.

If $b$ is in M(32,51,31), then $B$ is in M(32,51,23) or M(32,51,31).

## Projective indecomposable modules

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

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

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

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

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

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

## Irreducible characters

All irreducible characters have height zero.

## Cartan matrix

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

## Decomposition matrix

$\left( \begin{array}{ccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 3 & 2 & 2 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 4 & 1 & 3 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 2 & 1 & 0 & 1 & 0 & 1 \end{array}\right)$