# M(9,2,4)

M(9,2,4) - $k(S_3 \times S_3)$
Representative: $k(S_3 \times S_3)$ $C_3 \times C_3$ $C_2 \times C_2$ 9 4 1 $\left( \begin{array}{cccc} 4 & 2 & 1 & 2 \\ 2 & 4 & 2 & 1 \\ 1 & 2 & 4 & 2 \\ 2 & 1 & 2 & 4 \\ \end{array} \right)$ Yes $\mathcal{O} (S_3 \times S_3)$ $\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ \end{array}\right)$ 1 $C_2 \wr C_2$[1] No No No

## Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,4>, d:<4,1>, e:<1,4>, f:<4,3>, g:<3,2>, h:<2,1>

Relations w.r.t. $k$: $ab=ef$, $bc=he$, $cd=gh$, $da=fg$, $aha=ede=0$, $bgb=hah=0$, $cfc=gbg=0$, $ded=fcf=0$

## Projective indecomposable modules

Labelling the simple $B$-modules by $1,2,3,4$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{cccc} \begin{array}{ccccc} & & 1 & & \\ & 2 & & 4 & \\ 1 & & 3 & & 1 \\ & 4 & & 2 & \\ & & 1 & & \\ \end{array}, & \begin{array}{ccccc} & & 2 & & \\ & 1 & & 3 & \\ 2 & & 4 & & 2 \\ & 3 & & 1 & \\ & & 2 & & \\ \end{array}, & \begin{array}{ccccc} & & 3 & & \\ & 2 & & 4 & \\ 3 & & 1 & & 3 \\ & 4 & & 2 & \\ & & 3 & & \\ \end{array}, & \begin{array}{ccccc} & & 4 & & \\ & 1 & & 3 & \\ 4 & & 2 & & 4 \\ & 3 & & 1 & \\ & & 4 & & \\ \end{array} \\ \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. Proposition 4.3 of [BKL18]