The structure of generalized permanent semigroups (Q2365705)

Let \(M\) be the multiplicative semigroup of all \(n\times n\) matrices \(A=(a_{ij})\) over a field \(F,D\) the subsemigroup of diagonal matrices and \(H\) a subgroup of the symmetric group of degree \(n\). The authors introduce the generalized permanent function \(d_ H(A)=\sum_{\sigma\in H}a_{1\sigma(1)}\cdot\dots\cdot a_{n\sigma(n)}\). A matrix semigroup \(S\) such that \(D\subseteq S\subseteq M\) is called a \(d_ H\)-semigroup if \(d_ H(AB)=d_ H(A)d_ H(B)\) for all \(A,B\in S\). The authors describe, in considerable detail and under mild restrictions on \(n\) and \(F\), the maximal \(d_ H\)-semigroups. A prominent part in the proofs is taken by certain \(d_ H\)-semigroups \(S(K)\) corresponding to the subgroups \(K\) of \(H\). One consequence of the results is that, if \(A\) is an element of a \(d_ H\)-semigroup, then at most one term \(a_{1\sigma(1)}\cdot\dots\cdot a_{n\sigma(n)}\) in \(d_ H(A)\) is nonzero.