Spaces of generators for matrix algebras with involution

Abstract

Let $k$ be an algebraically closed field of characteristic different from 2. Up to isomorphism, the algebra $\operatorname{Mat}_{n \times n}(k)$ can be endowed with a $k$-linear involution in one way if $n$ is odd and in two ways if $n$ is even. In this paper, we consider $r$-tuples $A_\bullet \in \operatorname{Mat}_{n\times n}(k)^r$ such that the entries of $A_\bullet$ fail to generate $\operatorname{Mat}_{n\times n}(k)$ as an algebra with involution. We show that the locus of such $r$-tuples forms a closed subvariety $Z(r;V)$ of $\operatorname{Mat}_{n\times n}(k)^r$ that is not irreducible. We describe the irreducible components and we calculate the dimension of the largest component of $Z(r;V)$ in all cases. This gives a numerical answer to the question of how generic it is for an $r$-tuple $(a_1, \dots, a_r)$ of elements in $\operatorname{Mat}_{n\times n}(k)$ to generate it as an algebra with involution.