Permanent groups

Abstract:

A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. Let $A \circ B$ denote the Hadamard product of matrices A and B. The set of groups G of nonsingular $n \times n$ matrices which contain the diagonal group $\mathcal {D}$ and such that for every pair A, B of matrices in G we have $A \circ {B^T} \in \mathcal {D}$ is denoted by ${\mathcal {A}_n}$. If the underlying field has at least three elements then ${\mathcal {A}_n}$ consists of permanent groups. A partial converse is available: If a permanent group G is generated by $\mathcal {D}$ together with a set S of elementary matrices and a set Q of permutation matrices then $G = HK$ where H is the subgroup generated by Q and K is generated by $\mathcal {D}$ and S, and $K \in {\mathcal {A}_n}$.