Difference between revisions of "M(2^n,1,1)"

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

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>

Relations w.r.t. $k$: $a^{2^n}=0$

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 $B$ or $b$ is in M(2^n,1,1), then $B$ and $b$ must be Morita equivalent.

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.