Difference between revisions of "M(8,3,1)"

M(8,3,1) - $kD_8$
Representative: $kD_8$ $D_8$ $1$ 5 1 1 $\left( \begin{array}{c} 8 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O}D_8$ $\left( \begin{array}{c} 2 \\ 1 \\ 1 \\ 1 \\ 1 \\ \end{array}\right)$ 1 {{{PIgroup}}} No Yes Forms a derived equivalence class Yes {{{pcoveringblocks}}}

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>, b:<1,1>

Relations w.r.t. $k$: a^2=b^2=(ab)^2+(ba)^2=0

Projective indecomposable modules

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

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

Irreducible characters

$k_0(B)=4, k_1(B)=1$