On linear preservers of permanental rank

• Published in: arXiv.org 2023-10
• Main Authors: Guterman, Alexander, Spiridonov, Igor
• Format: Article
• Language: English
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2308.14526

Let \({\rm Mat}_n(\mathbb{F})\) denote the set of square \(n\times n\) matrices over a field \(\mathbb{F}\) of characteristic different from two. The permanental rank \({\rm prk}\,(A)\) of a matrix \(A \in{\rm Mat}_{n}(\mathbb{F})\) is the size of the maximal square submatrix in \(A\) with nonzero permanent. By \(\Lambda^{k}\) and \(\Lambda^{\leq k}\) we denote the subsets of matrices \(A \in {\rm Mat}_{n}(\mathbb{F})\) with \({\rm prk}\,(A) = k\) and \({\rm prk}\,(A) \leq k\), respectively. In this paper for each \(1 \leq k \leq n-1\) we obtain a complete characterization of linear maps \(T: {\rm Mat}_{n}(\mathbb{F}) \to {\rm Mat}_{n}(\mathbb{F})\) satisfying \(T(\Lambda^{\leq k}) = \Lambda^{\leq k}\) or bijective linear maps satisfying \(T(\Lambda^{\leq k}) \subseteq \Lambda^{\leq k}\). Moreover, we show that if \(\mathbb{F}\) is an infinite field, then \(\Lambda^{k}\) is Zariski dense in \(\Lambda^{\leq k}\) and apply this to describe such bijective linear maps satisfying \(T(\Lambda^{k}) \subseteq \Lambda^{k}\).