M(5,1,2)

M(5,1,2) - $kD_{10}$
Representative: $kD_{10}$ $C_5$ $C_2$ 4 2 1 $\left( \begin{array}{cc} 3 & 2 \\ 2 & 3 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} D_{10}$ $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \\ \end{array}\right)$ 1 $\mathcal{T}(B)=C_4$ {{{PIgroup}}} Yes $kD_{10}$ Yes M(5,1,3) Yes {{{coveringblocks}}} {{{coveredblocks}}} {{{pcoveringblocks}}}

Basic algebra

Quiver: a:<1,2>, b:<2,1>

Relations w.r.t. $k$: ababa=babab=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(5,1,2), then $B$ must lie in M(5,1,1), M(5,1,2) or M(5,1,4).

If $B$ lies in M(5,1,2), then $b$ must lie in M(5,1,1), M(5,1,2) or M(5,1,4). Examples needed.

Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, S_2$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{cc} \begin{array}{c} S_1 \\ S_2 \\ S_1 \\ S_2 \\ S_1 \\ \end{array}, & \begin{array}{c} S_2 \\ S_1 \\ S_2 \\ S_1 \\ S_2 \\ \end{array} \end{array}$

Irreducible characters

All irreducible characters have height zero.