Irredundant generating sets for matrix algebras

Abstract

Let F be a field. We show that the largest irredundant generating sets for the algebra of $n\times n$ matrices over F have $2n-1$ elements when $n>1$. (A result of Laffey states that the answer is $2n-2$ when $n>2$, but its proof contains an error.) We further give a classification of the largest irredundant generating sets when $n\in \{2,3\}$ and F is algebraically closed. We use this description to compute the dimension of the variety of $(2n-1)$-tuples of $n\times n$ matrices which form an irredundant generating set when $n\in \{2,3\}$, and draw some consequences to locally redundant generation of Azumaya algebras. In the course of proving the classification, we also determine the largest sets S of subspaces of $F^3$ with the property that every $V\in S$ admits a matrix stabilizing every subspace in $S-\left\{V\right\}$ and not stabilizing V.