# M(8,3,5)

M(8,3,5) - $B_0(kA_7)$
Representative: $B_0(kA_7)$ $D_8$ $1$ 5 3 1 $\left( \begin{array}{ccc} 3 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 2 \\ \end{array} \right)$ Yes Yes No $\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \\ \end{array}\right)$ 1 {{{PIgroup}}} No Yes M(8,3,4), M(8,3,6) No {{{pcoveringblocks}}}

These are tame blocks, and appear in the family $D(3 {\cal B})_1$ in Erdmann's classification (see [Er87]). Derived equivalences over $k$ are established in [Li94b].

## Basic algebra

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

Relations w.r.t. $k$: $ab=ea=be=dc=0$, $cdeb=ebcd$, $a^2=bcde$

## Projective indecomposable modules

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

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

## Irreducible characters

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