Consider a commutative ring $x_ix_j = N_{ij}^k x_k$, where $N_{ij}^k \in\{0,1,2,3,\cdots\}$. (This is actually a fusion ring)

How to find the irreducible integer-matrix representations of the above ring? (ie what is the algorithm to find the irreducible representations given $N_{ij}^k$.)