Linear transformations preserving the Grassmannian over \(\mathcal M_n(\mathbb Z_+)\)
Let \({\mathbb Z}_+\) be the set of non-negative integers and \({\mathcal M}_n({\mathbb Z}_+)\) denote the set of \(n \times n\) matrices with coefficients from \({\mathbb Z}_+\). The semiring \({\mathbb Z}_+\) is an example of an antinegative semiring, i. e., one for which \(a+b=0\) implies that \(a=b=0\). The problem of determining the linear operators on the \(n \times n\) matrix algebra \({\mathcal M}_n({\mathbb F})\) over a field \({\mathbb F}\) that leave certain matrix relations invariant is widely studied. Here, a characterization of linear transformations on matrices with non-negative integer coefficients that preserve the matrix relation \(XYZ+ZYX=YXZ+ZXY\) is obtained. The polynomial \(XYZ+ZYX=YXZ+ZXY\) is known as a Grassmannian since it is identically zero if evaluated over the infinite dimensional Grassmann algebra [\textit{D. Krakowski} and \textit{A. Regev}, Trans. Am. Math. Soc. 181, 429--438 (1973; Zbl 0289.16015)].
scientific article; zbMATH DE number 2126340
scientific article