M(9,1,1)

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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 $\mathcal{L}(B)=C_9:C_6$ Yes $kC_9$ Yes Forms a derived equivalence class Yes

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>

Relations w.r.t. $k$: a^9=0

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.