M(27,1,1)

M(27,1,1) - $kC_{27}$
Representative: $kC_{27}$ $C_{27}$ $1$ 27 1 1 $\left( \begin{array}{c} 27 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} C_{27}$ $\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}\right)$ 1 $\mathcal{L}(B)=C_{27}:C_{18}$ {{{PIgroup}}} Yes $kC_{27}$ 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^{27}=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(27,1,1), then $B$ must lie in M(27,1,1) or M(27,1,2). For example consider the principal blocks of $C_{27} \triangleleft D_{18}$.

If $B$ lies in M(27,1,1), then $b$ must lie in M(27,1,1) or M(27,1,2). 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.