# M(16,3,3)

M(16,3,3) - $k(A_4:C_4)$[1]
[[File:|250px]]
Representative: $k(A_4:C_4)$ MNA(2,1) $1$ 10 2 1 $\left( \begin{array}{cc} 8 & 4 \\ 4 & 6 \\ \end{array} \right)$ $\left( \begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ 1 & 1 \\ \end{array}\right)$[2]

## Basic algebra

Quiver: a: <1,2>, b:=<1,1>, c:=<2,1>, d:=<2,2>, e:=<2,2>

Relations w.r.t. $k$:

## Projective indecomposable modules

Labelling the simple $B$-modules by $1, 2$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{cc} \begin{array}{c} 1 \\ 1 \ 2 \\ 1 \ 1 \ 2 \\ 1 \ 1 \ 2 \\ 1 \ 2 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 1 \ 2 \ 2 \\ 1 \ 2 \\ 1 \ 2 \\ 1 \\ 2 \\ \end{array} \\ \end{array}$

## Irreducible characters

$k_0(B)=8$, $k_1(B)=2$

## Notes

1. $A_4:C_4$ is SmallGroup(48,30).
2. This is the only possible decomposition matrix with the given Cartan matrix and $k(B)=10$.