# M(3,1,2)

M(3,1,2) - $kS_3$
Representative: $kS_3$ $C_3$ $C_2$ 3 2 1 $\left( \begin{array}{cc} 2 & 1 \\ 1 & 2 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} S_3$ $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 1 \\ \end{array}\right)$ 1 $\mathcal{T}(B)=C_2$ {{{PIgroup}}} Yes $kS_3$ Yes M(3,1,1) Yes M(3,1,1) M(3,1,1) {{{pcoveringblocks}}}

These are very frequently occuring blocks with cyclic defect groups, so are described in work culminating in [Li96] .

## Basic algebra

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

Relations w.r.t. $k$: aba=bab=0

## Covering blocks and covered blocks

Let $N \triangleleft G$ with $p'$-index and let $B$ be a block of $\mathcal{O} G$ covering a block $b$ of $\mathcal{O} N$.

If $b$ lies in M(3,1,2), then $B$ must lie in M(3,1,1) or M(3,1,2). Example needed.

If $B$ lies in M(3,1,2), then $b$ must lie in M(3,1,1) or M(3,1,2). For example consider the principal blocks of $C_3 \triangleleft S_3$.

## Projective indecomposable modules

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

$\begin{array}{cc} \begin{array}{c} S_1 \\ S_2 \\ S_1 \\ \end{array}, & \begin{array}{c} S_2 \\ S_1 \\ S_2 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.