# M(9,2,3)

M(9,2,3) - $k((C_3 \times C_3):C_2)$
Representative: $k((C_3 \times C_3):C_2)$ $C_3 \times C_3$ $C_2$ 6 2 1 $\left( \begin{array}{cc} 5 & 4 \\ 4 & 5 \\ \end{array} \right)$ Yes Yes No $\mathcal{O} ((C_3 \times C_3):C_2)$ $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ \end{array}\right)$ 1 $C_2$[1] No No No

The representative $k((C_3 \times C_3):C_2)$ is given by $C_3 \times C_3$ acted on by an element inverting those of $C_3 \times C_3$, i.e., it is the group SmallGroup(18,4).

## Basic algebra

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

Relations w.r.t. $k$: $ad=bc$, $cb=da$, $aca=bdb=0$, $cac=dbd=0$

## Projective indecomposable modules

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

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

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. Proposition 4.3 of [BKL18]