# M(8,5,8)

M(8,5,8) - $B_0(k(\rm Aut (SL_2(8))))$
Representative: $B_0(k(\rm Aut (SL_2(8))))$ $C_2 \times C_2 \times C_2$ $C_7:C_3$ 8 5 1 $\left( \begin{array}{ccccccc} 4 & 2 & 2 & 4 & 2 \\ 2 & 4 & 2 & 4 & 2 \\ 2 & 2 & 4 & 4 & 2 \\ 4 & 4 & 4 & 8 & 3 \\ 2 & 2 & 2 & 3 & 2 \\ \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O}(\rm Aut (SL_2(8))))$ $\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ \end{array}\right)$[1] 1 $C_3$[2] No Yes M(8,5,6), M(8,5,7)[3] Yes [4] Potentially M(8,5,5) M(8,5,5)

The principal 2-blocks of all Ree groups ${}^2G_2(3^{2m+1})$ belong to this Morita equivalence class.

## Basic algebra

Quiver: a:<4,1>, b:<1,4>, c:<2,4>, d:<4,2>, e:<4,3>, f:<3,4>, g:<4,5>, h:<5,4>[5]

Relations w.r.t. $k$:

## Other notatable representatives

${\rm Aut (SL_2(8))} \cong {}^2G_2(3)$, and the blocks $B_0(\mathcal{O}({}^2G_2(3^{2m+1})))$ are Morita equivalent for all $m$. This follows from Example 3.3 of [Ok97], where, as noted in 6.2.2 of [CR13] the Morita equivalence is splendid and so lifts to $\mathcal{O}$

By [Ea16] the principal block of each subgroup of ${\rm Aut}({}^2G_2(3^{2m+1}))$ containing ${}^2G_2(3^{2m+1})$ is in this Morita equivalence class.

## Projective indecomposable modules

Labelling the simple $B$-modules by $1,2,3,4,5$, the projective indecomposable modules have Loewy structure as follows[6]:

$\begin{array}{ccccc} \begin{array}{c} 1 \\ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \\ 3 \\ \end{array}, & \begin{array}{c} 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \ 4 \ 4 \\ 1 \ 1 \ 2 \ 2 \ 3 \ 3 \ 5 \\ 4 \ 4 \ 4 \\ 1 \ 2 \ 3 \ 5 \\ 4 \\ \end{array}, & \begin{array}{c} 5 \\ 4 \\ 1 \ 2 \ 3 \\ 4 \\ 1 \ 2 \ 3 \\ 4 \\ 5 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. Decomposition matrix taken from [1], although it was first determined in [LM80] following partial results of Fong
2. See [EL18c]
3. Derived equivalent by [GO97]
4. As noted in [CR13] the derived equivalences in [GO97] are splendid and so lift to $\mathcal{O}$
5. Computed using MAGMA
6. The structure of the projective indecomposable modules was first given in [LM80], although with a mistake corrected in [GO97]