# M(8,3,3)

M(8,3,3) - $kS_4$
Representative: $kS_4$ $D_8$ $1$ 5 2 1 $\left( \begin{array}{cc} 4 & 2 \\ 2 & 3 \\ \end{array} \right)$ Yes Yes No $\left( \begin{array}{c} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \\ \end{array}\right)$ {{{PIgroup}}} No Yes M(8,3,2) {{{pcoveringblocks}}}

These are tame blocks, and appear in the family $D(2 {\cal B})$ in Erdmann's classification (see [Er87] ). Derived equivalences over $k$ are established in [Ho97].

## Basic algebra

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

Relations w.r.t. $k$: $a^2=bd=dc=cb=0$, $abc=bca$, $cab=d^2$

## Projective indecomposable modules

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

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

## Irreducible characters

$k_0(B)=4, k_1(B)=1$