# M(9,2,2)

M(9,2,2) - $k(S_3 \times C_3)$
Representative: $k(S_3 \times C_3)$ $C_3 \times C_3$ $C_2$ 9 2 1 $\left( \begin{array}{cc} 6 & 3 \\ 3 & 6 \\ \end{array} \right)$ Yes Yes No $\mathcal{O} (C_3 \times C_3)$ $\left( \begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ \end{array}\right)$ 1 No No none known No

## Basic algebra

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

Relations w.r.t. $k$: $a^3=d^3=0$, $ab=bd$, $ca=dc$, $bcb=cbc=0$

## Projective indecomposable modules

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

$\begin{array}{cc} \begin{array}{ccccc} & & 1 & & \\ & 1 & & 2 & \\ 1 & & 2 & & 1 \\ & 2 & & 1 & \\ & & 1 & & \\ \end{array}, & \begin{array}{ccccc} & & 2 & & \\ & 2 & & 1 & \\ 2 & & 1 & & 2 \\ & 1 & & 2 & \\ & & 2 & & \\ \end{array} \\ \end{array}$

## Irreducible characters

All irreducible characters have height zero.