# M(3^n,1,3)

M(3^n,1,3) - $B_0(kPSL_2(2^{(3^{n-1})}))$
Representative: $B_0(kPSL_2(2^{(3^{n-1})}))$ $C_{3^n}$ $C_2$ $\frac{3^n+3}{2}$ 2 1 $\left( \begin{array}{cc} 2 & 1 \\ 1 & \frac{3^n+1}{2} \\ \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O} PSL_2(2^{(3^{n-1})}))$ $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ \vdots & \vdots \\ 0 & 1 \\ 1 & 1 \\ \end{array}\right)$ 1 {{{PIgroup}}} Yes Yes M(3^n,1,2) Yes {{{coveringblocks}}} {{{coveredblocks}}} {{{pcoveringblocks}}}

## Basic algebra

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

Relations w.r.t. $k$: $ac=cb=ba-c^{(3^n-1)/2}=0$

## Other notatable representatives

$B_0(kPSL_2(q_n))$ for any $q_n$ a prime power such that $(q_n+1)_3=3^n$.

## Projective indecomposable modules

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

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

## Irreducible characters

All irreducible characters have height zero.