Irredundant Generating Sets for Matrix Algebras
Yonatan Blumenthal, Uriya First
TL;DR
This work establishes that the largest irredundant generating sets of $ ext{M}_n(F)$ have size $2n-1$ for $n>1$, correcting a previous bound due to an error in Laffey's proof and providing a constructive example. It then classifies the maximal irredundant generating sets for $n=2,3$ over algebraically closed fields, linking the problem to $ ext{GL}_n(F)$-independence of subspaces and deriving exact dimensions for corresponding irredundant-tuples varieties. The 5-element case for $n=3$ is fully described up to simple transformations, with key dimensions $ ext{dim } I_5^{(3)}=19$ and implications for the geometry of matrix tuples. An application to Azumaya algebras shows that, under certain dimension hypotheses, degree-3 Azumaya algebras over $R$ admit locally redundant 5-tuples generating them, illustrating a bridge between linear algebra, algebraic geometry, and noncommutative algebra.
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 $V$.