# M(8,4,2)

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M(8,4,2) - $B_0(kSL_2(5))$
Representative: $B_0(kSL_2(5))$ $Q_8$ $C_3$ 7 3 1 $\left( \begin{array}{ccc} 4 & 4 & 2 \\ 4 & 8 & 4 \\ 2 & 4 & 4 \\ \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O}SL_2(5))$ $\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ 1 & 2 & 1 \\ \end{array}\right)$ 1 {{{PIgroup}}} No Yes M(8,4,3) No

These are tame blocks, and appear in the family $D(3 {\cal A})_2$ in Erdmann's classification (see [Er88a], [Er88b]). The class lifts to a unique $\mathcal{O}$-Morita equivalence class by [Ei16]. A derived equivalence with M(8,4,3) over $k$ was established in [Ho97].

## Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,2>, d:<2,1>

Relations w.r.t. $k$: ada=abcdabc, dad=bcdabcd, cbc=cdabcda, bcb=dabcdab, adab=cbcd=0

## Projective indecomposable modules

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

$\begin{array}{ccc} \begin{array}{ccc} & 1 & \\ & 2 & \\ \begin{array}{c} 1 \\ \end{array} & \oplus & \begin{array}{c} 3 \\ 2 \\ 1 \\ 2 \\ 3 \\ \end{array} \\ & 2 & \\ & 1 & \\ \end{array}, & \begin{array}{c} 2 \\ 1 \ 3 \\ 2 \ 2 \\ 3 \ 1 \\ 2 \ 2 \\ 1 \ 3 \\ 2 \ 2 \\ 3 \ 1 \\ 2 \\ \end{array}, & \begin{array}{ccc} & 3 & \\ & 2 & \\ \begin{array}{c} 3 \\ \end{array} & \oplus & \begin{array}{c} 1 \\ 2 \\ 3 \\ 2 \\ 1 \\ \end{array} \\ & 2 & \\ & 3 & \\ \end{array} \end{array}$

## Irreducible characters

$k_0(B)=4, k_1(B)=3$