# M(4,2,1)

M(4,2,1) - $k(C_2 \times C_2)$
Representative: $k(C_2 \times C_2)$ $C_2 \times C_2$ $1$ 4 1 1 $(k \times k):GL_2(k)$ $\left( \begin{array}{c} 4 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} (C_2 \times C_2)$ $\left( \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \\ \end{array}\right)$ 1 $\mathcal{L}(B)=S_4$ $S_4 \times C_2$ Yes $k(C_2 \times C_2)$ Yes Forms a derived equivalence class Yes M(4,2,1), M(4,2,3) (complete) M(4,2,1), M(4,2,3)[1] (complete) M(8,2,1), M(8,3,1), M(8,5,1) (complete)

These are nilpotent blocks.

## Basic algebra

Quiver: a:<1,1>, b:<1,1>

Relations w.r.t. $k$: a^2=b^2=ab+ba=0

## Other notatable representatives

Block number 2 of $k PGL_3(7)$ in the labelling used in [2]

## Projective indecomposable modules

Labelling the unique simple $B$-module by $S_1$, the unique projective indecomposable module has Loewy structure as follows:

$\begin{array}{ccc} & S_1 & \\ S_1 & & S_1 \\ & S_1 & \\ \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. For example consider block number 2 of $PSL_3(7) \triangleleft PGL_3(7)$ in the labelling used in [1].