# M(8,3,4)

M(8,4,3) - $B_0(kPSL_2(9))$
Representative: $B_0(kPSL_2(9))$ $D_8$ $1$ 5 3 1 $\left( \begin{array}{ccc} 3 & 4 & 2 \\ 4 & 8 & 4 \\ 2 & 4 & 3 \\ \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O}A_5)$ $\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 1 \\ \end{array}\right)$ 1 {{{PIgroup}}} No Yes M(8,3,5), M(8,3,6) {{{pcoveringblocks}}}

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

## Basic algebra

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

Relations w.r.t. $k$: $ad=cb=(bcda)^2+(dabc)^2=0$

## Projective indecomposable modules

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

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

## Irreducible characters

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