# M(5,1,4)

M(5,1,4) - $k(C_5:C_4)$
Representative: $k(C_5:C_4)$ $C_5$ $C_4$ 5 4 1 $\left( \begin{array}{cccc} 2 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O}(C_5:C_4)$ $\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ \end{array}\right)$ 1 Yes Yes M(5,1,5), M(5,1,6) Yes {{{coveringblocks}}} {{{coveredblocks}}}

## Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,4>, d:<4,1>

Relations w.r.t. $k$: abcda=bcdab=cdabc=dabcd=0

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, S_2, S_3, S_4$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{cccc} \begin{array}{c} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_1 \\ \end{array}, & \begin{array}{c} S_2 \\ S_3 \\ S_4 \\ S_1 \\ S_2 \\ \end{array}, & \begin{array}{c} S_3 \\ S_4 \\ S_1 \\ S_2 \\ S_3 \\ \end{array}, & \begin{array}{c} S_4 \\ S_1 \\ S_2 \\ S_3 \\ S_4 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.