# M(3^n,1,2)

M(3^n,1,2) - $kD_{2.3^n}$
Representative: $kD_{2.3^n}$ $C_{3^n}$ $C_2$ $\frac{3^n+3}{2}$ 2 1 $\left( \begin{array}{cc} \frac{3^n+1}{2} & \frac{3^n-1}{2} \\ \frac{3^n-1}{2} & \frac{3^n+1}{2} \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} D_{18}$ $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 1 \\ \vdots & \vdots \\ 1 & 1 \\ 1 & 1 \\ \end{array}\right)$ 1 {{{PIgroup}}} Yes $kD_{2.3^n}$ Yes M(3^n,1,3) Yes {{{coveringblocks}}} {{{coveredblocks}}} {{{pcoveringblocks}}}

## Basic algebra

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

Relations w.r.t. $k$: $a(ba)^{(3^n-1)/2}=b(ab)^{(3^n-1)/2}=0$

## Projective indecomposable modules

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

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

## Irreducible characters

All irreducible characters have height zero.