# M(4,2,3)

M(4,2,3) - $kA_4$
Representative: $kA_4$ $C_2 \times C_2$ $C_3$ 4 3 1 $(k^* \times k^* \times C_3):C_2$[1] $\left( \begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O}A_4$ $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ \end{array}\right)$ 1 $\mathcal{T}(B)=S_3$[2] $S_4 \times C_2$[3] Yes $\mathcal{O}A_4$ Yes M(4,2,2) Yes M(4,2,1)[4], M(4,2,3) (complete) M(4,2,1), M(4,2,2) (complete) M(8,3,3), M(8,5,3) (complete)[5]

## Basic algebra

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

Relations w.r.t. $k$: ab=bc=ca=0, df=fe=ed=0, ad=fc, be=da, cf=eb

## Other notatable representatives

Block number 2 of $k PSL_3(7)$ in the labelling used in [2]

## Projective indecomposable modules

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

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

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. This is an elementary calculation.
2. Every Morita equivalence is a source algebra equivalence by [CEKL13], so ${\rm Pic}(B)=\mathcal{T}(B)$
3. See, for example, Proposition 2.8 of [EL18a]
4. For example consider block number 2 of $PSL_3(7) \triangleleft PGL_3(7)$ in the labelling used in [1]. The covering block of $PGL_3(7)$ is nilpotent.
5. A covering block cannot have defect group $C_4 \times C_2$.