On linear preservers of permanental rank

Abstract

Let Matn(F) denote the set of square n×n matrices over a field F of characteristic different from two. The permanental rank prk(A) of a matrix A∈Matn(F) is the size of the maximal square submatrix in A with nonzero permanent. By Λk and Λ≤k we denote the subsets of matrices A∈Matn(F) with prk(A)=k and prk(A)≤k, respectively. In this paper for each 1≤k≤n−1 we obtain a complete characterization of linear maps T:Matn(F)→Matn(F) satisfying T(Λ≤k)=Λ≤k or bijective linear maps satisfying T(Λ≤k)⊆Λ≤k. Moreover, we show that if F is an infinite field, then Λk is Zariski dense in Λ≤k and apply this to describe such bijective linear maps satisfying T(Λk)⊆Λk.