# M(4,2,2)

M(4,2,2) - $B_0(kA_5)$
Representative: $B_0(kA_5)$ $C_2 \times C_2$ $C_3$ 4 3 1 $k^* \wr C_2$[1] $\left( \begin{array}{ccc} 2 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 2 \\ \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O}A_5)$ $\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right)$ 1 $\mathcal{T}(B)=C_2$[2] $S_4 \times C_2$[3] Yes $B_0(\mathcal{O}A_5)$ Yes M(4,2,3) Yes M(4,2,2) (complete) M(4,2,2) (complete) M(8,3,2), M(8,5,2) (complete)[4]

## Basic algebra

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

Relations w.r.t. $k$: ad=cb=bcda+dabc=0

## Projective indecomposable modules

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

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

## 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. This follows from the result for M(4,2,3) since blocks in these classes are derived equivalent and so perfectly isometric
4. A covering block cannot have defect group $C_4 \times C_2$.