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 Zariski-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 $S-\{V\}$ and not stabilizing $V$.