# M(8,3,2)

M(8,3,2) - $B_0(kPGL_2(5))$
Representative: $B_0(kPGL_2(5))$ $D_8$ $1$ 5 2 1 $\left( \begin{array}{cc} 3 & 4 \\ 4 & 8 \\ \end{array} \right)$ Yes Yes No $\left( \begin{array}{c} 0 & 1 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 2 \\ \end{array}\right)$ {{{PIgroup}}} No Yes M(8,3,3) {{{pcoveringblocks}}}

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

## Basic algebra

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

Relations w.r.t. $k$: $ab=c^2=0$, $(cba)^2=(bac)^2$

## Projective indecomposable modules

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

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

## Irreducible characters

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