# M(8,5,4)

M(8,5,4) - $k((C_2 \times C_2 \times C_2):C_7)$
Representative: $k((C_2 \times C_2 \times C_2):C_7)$ $C_2 \times C_2 \times C_2$ $C_7$ 8 7 1 $\left( \begin{array}{ccccccc} 2 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 2 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 2 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 2 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} ((C_2 \times C_2 \times C_2):C_7)$ $\left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array}\right)$ 1 $C_7 : C_3$[1] $S_7 \times C_2$[2] No Yes M(8,5,5) Yes

## Basic algebra

Quiver: $a_1:\lt 1,2\gt , a_2:\lt 2,3\gt$, $a_3:\lt 3,4\gt , a_4:\lt 4,5\gt$, $a_5:\lt 5,6\gt , a_6:\lt 6,7\gt , a_7:\lt 7,1\gt$, $b_1:\lt 1,3\gt , b_2:\lt 2,4\gt$, $b_3:\lt 3,5\gt , b_4:\lt 4,6\gt$, $b_5:\lt 5,7\gt , b_6:\lt 6,1\gt , b_7:\lt 7,2\gt$, $c_1:\lt 1,5\gt , c_2:\lt 2,6\gt$, $c_3:\lt 3,7\gt , c_4:\lt 4,1\gt$, $c_5:\lt 5,2\gt , c_6:\lt 6,3\gt , c_7:\lt 7,4\gt$[3]

Relations w.r.t. $k$: $a_1a_2=a_2a_3=a_3a_4=a_4a_5=a_5a_6=a_6a_7=a_7a_1=0$, $b_1b_3=b_2b_4=b_3b_5=b_4b_6=b_5b_7=b_6b_1=b_7b_2=0$, $c_1c_5=c_2c_6=c_3c_7=c_4c_1=c_5c_2=c_6c_3=c_7c_4=0$, $a_1b_2=b_1a_3$, $a_2b_3=b_2a_4$, $a_3b_4=b_3a_5$, $a_4b_5=b_4a_6$, $a_5b_6=b_5a_7$, $a_6b_7=b_6a_1$, $a_7b_1=b_7a_2$, $a_1c_2=c_1a_5$, $a_2c_3=c_2a_6$, $a_3c_4=c_3a_7$, $a_4c_5=c_4a_1$, $a_5c_6=c_5a_2$, $a_6c_7=c_6a_3$, $a_7c_1=c_7a_4$, $b_1c_3=c_1b_5$, $b_2c_4=c_2b_6$, $b_3c_5=c_3b_7$, $b_4c_6=c_4b_1$, $b_5c_7=c_5b_2$, $b_6c_1=c_6b_3$, $b_7c_2=c_7b_4$ $a_1b_2c_4=b_1c_3a_7=c_1a_5b_6$ $a_2b_3c_5=b_2c_4a_1=c_2a_6b_7$ $a_3b_4c_6=b_3c_5a_2=c_3a_7b_1$ $a_4b_5c_7=b_4c_6a_3=c_4a_1b_2$ $a_5b_6c_1=b_5c_7a_4=c_5a_2b_3$ $a_6b_7c_2=b_6c_1a_5=c_6a_3b_4$ $a_7b_1c_3=b_7c_2a_6=c_7a_4b_5$

## Projective indecomposable modules

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

$\begin{array}{ccccccc} \begin{array}{c} S_1 \\ S_2 S_3 S_5 \\ S_4 S_6 S_7 \\ S_1 \\ \end{array}, & \begin{array}{ccc} S_2 \\ S_3 S_4 S_6 \\ S_1 S_5 S_7 \\ S_2 \\ \end{array}, & \begin{array}{ccc} S_3 \\ S_4 S_5 S_7 \\ S_1 S_2 S_6 \\ S_3 \\ \end{array} , & \begin{array}{ccc} S_4 \\ S_5 S_6 S_1 \\ S_2 S_3 S_7 \\ S_4 \\ \end{array}, & \begin{array}{ccc} S_5 \\ S_6 S_7 S_2 \\ S_1 S_3 S_4 \\ S_3 \\ \end{array}, & \begin{array}{ccc} S_6 \\ S_7 S_1 S_3 \\ S_2 S_4 S_5 \\ S_6 \\ \end{array}, & \begin{array}{ccc} S_7 \\ S_1 S_2 S_4 \\ S_3 S_5 S_6 \\ S_7 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. See [BKL18]
2. Using GAP programme from [Ru11].
3. Thanks to Malka Schaps for showing us this way of drawing the quiver, from [SSS98].