# M(9,1,1)

M(9,1,1) - $kC_9$
Representative: $kC_9$ $C_9$ $1$ 9 1 1 $\left( \begin{array}{c} 9 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} C_9$ $\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}\right)$ 1 $\mathcal{L}(B)=C_9:C_6$ {{{PIgroup}}} Yes $kC_9$ Yes Forms a derived equivalence class Yes {{{coveringblocks}}} {{{coveredblocks}}} {{{pcoveringblocks}}}

These are nilpotent blocks.

## Basic algebra

Quiver: a:<1,1>

Relations w.r.t. $k$: a^9=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(9,1,1), then $B$ must lie in M(9,1,1) or M(9,1,2). For example consider the principal blocks of $C_9 \triangleleft D_{18}$.

If $B$ lies in M(9,1,1), then $b$ must lie in M(9,1,1) or M(9,1,2). Examples needed.

## Projective indecomposable modules

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

$\begin{array}{c} S_1 \\ S_1 \\ \vdots \\ S_1 \\ \end{array}$

## Irreducible characters

All irreducible characters have height zero.