# M(25,1,1)

M(25,1,1) - $kC_{25}$
Representative: $kC_{25}$ $C_{25}$ $1$ 25 1 1 $k^{23}:k^*$ $\left( \begin{array}{c} 25 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} C_{25}$ $\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}\right)$ 1 $\mathcal{L}(B)=C_{25}:C_{20}$ {{{PIgroup}}} Yes $kC_{25}$ Yes Forms a derived equivalence class Yes M(25,1,2), M(25,1,4) {{{pcoveringblocks}}}

These are nilpotent blocks.

## Basic algebra

Quiver: a:<1,1>

Relations w.r.t. $k$: $a^{25}=0$

## 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(25,1,1), then $B$ must lie in M(25,1,1), M(25,1,2) or M(25,1,4). For example consider the principal blocks of $C_{25} \triangleleft D_{50}, C_{25}:C_4$.

If $B$ lies in M(25,1,1), then $b$ must lie in M(25,1,1), M(25,1,2) or M(25,1,4). Example 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.