# M(8,5,2)

M(8,5,2) - $B_0(k(A_5 \times C_2))$
Representative: $B_0(k(A_5 \times C_2))$ $C_2 \times C_2 \times C_2$ $C_3$ 8 3 1 $\left( \begin{array}{ccc} 8 & 4 & 4 \\ 4 & 4 & 2 \\ 4 & 2 & 4 \\ \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O} (A_5 \times C_2))$ $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right)$ 1 $C_2 \times C_2$[1] $S_4 \times C_2 \times C_2$[2] No Yes M(8,5,2) Yes

## Basic algebra

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

Relations w.r.t. $k$: $ad=cb=0$, $bcda=dabc$, $e^2=f^2=g^2=0$, $af=ea$, $bg=fb$, $cf=gc$, $de=fd$

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, S_2, S_3$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccc} \begin{array}{c} S_1 \\ S_1 S_2 S_3 \\ S_2 S_3 S_1 S_1 \\ S_1 S_1 S_3 S_2 \\ S_3 S_2 S_1 \\ S_1 \\ \end{array}, & \begin{array}{c} S_2 \\ S_2 S_1 \\ S_1 S_3 \\ S_3 S_1 \\ S_1 S_2 \\ S_2 \\ \end{array}, & \begin{array}{c} S_3 \\ S_2 S_1 \\ S_1 S_2 \\ S_3 S_1 \\ S_1 S_3 \\ S_3 \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

Back to $C_2 \times C_2 \times C_2$
1. ${\rm Pic}_{\mathcal{O}}(B_0(\mathcal{O}(C_2 \times A_5)))=\mathcal{L}(B_0(\mathcal{O}(C_2 \times A_5)))=\mathcal{L}(\mathcal{O}C_2) \times \mathcal{T}(B_0(\mathcal{O}A_5))$ by [EL18c], giving the isomorphism type of ${\rm Pic}_\mathcal{O}(B)$ in general.
2. By Theorem 3.7 of [EL18c].