# M(8,5,5)

M(8,5,5) - $B_0(kSL_2(8))$
Representative: $B_0(kSL_2(8))$ $C_2 \times C_2 \times C_2$ $C_7$ 8 7 1 $\left( \begin{array}{ccccccc} 8 & 4 & 4 & 4 & 2 & 2 & 2 \\ 4 & 4 & 2 & 2 & 0 & 2 & 1 \\ 4 & 2 & 4 & 2 & 1 & 0 & 2 \\ 4 & 2 & 2 & 4 & 2 & 1 & 0 \\ 2 & 0 & 1 & 2 & 2 & 0 & 0 \\ 2 & 2 & 0 & 1 & 0 & 2 & 0 \\ 2 & 1 & 2 & 0 & 0 & 0 & 2 \\ \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O}SL_2(8))$ $\left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ \end{array}\right)$[1] 1 $C_3$[2] {{{PIgroup}}} No Yes M(8,5,4) Yes[3] M(8,5,8) Potentially M(8,5,8) {{{pcoveringblocks}}}

The projective indecomposable modules of the $2$-blocks of the groups $SL_2(2^n)$ were computed by Alperin in [Al79] and the quiver and relations by Koshita in [Ko94]. A splendid derived equivalence with M(8,5,4) was constructed by Rouquier in [Ro95].

## Basic algebra

Quiver: a<1,2>, b:<2,3>, c:<3,2>, d:<2,1>, e:<1,4>, f:<4,5>, g:<5,4>, h:<4,1>, i:<1,6>, j:<6,7>, k:<7,6>, l:<6,1>

Relations w.r.t. $k$: $da=he=li=0$, $cb=gf=kj=0$, $bcd=dil$, $fgh=had$, $jkl=leh$, $abc=ila$, $efg=ade$, $ijk=ehi$, $cdi=gha=kle$, $lab=def=hij$

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. Decomposition matrix taken from [1]
2. See [EL18c]
3. A splendid Rickard equivalence is given in [Ro95, 2.3], which then lifts to $\mathcal{O}$