Spaces of generators for matrix algebras with involution

TL;DR

This paper investigates the structure of r-tuples of matrices over an algebraically closed field that fail to generate the full matrix algebra with involution, describing the geometry and dimensions of the associated algebraic varieties.

Contribution

It characterizes the locus of non-generating r-tuples as a reducible algebraic variety, describing its irreducible components and computing the dimension of its largest component.

Findings

The locus of non-generating r-tuples is a closed, reducible subvariety.

Explicit descriptions of irreducible components are provided.

The dimension of the largest component is calculated in all cases.

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.