Sweeping words and the length of a generic vector subspace of M_n(F)

TL;DR

This paper proves that for a generic set of matrices, words of a certain length span the entire matrix algebra, confirming a version of Paz's conjecture using combinatorial methods and directed multigraphs.

Contribution

It establishes a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras, using combinatorial graph constructions.

Findings

Words of degree 2d span M_n(F) when g^{2d} ≥ n^2

The proof employs generic matrices and directed multigraphs

Provides a combinatorial approach to a conjecture in matrix algebra

Abstract

The main result of this short note is a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras. Consider a generic g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever $g^{2d}\geq n^2$, the set of all words of degree 2d in A spans the full nxn matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of a special kind of directed multigraphs with few edge-disjoint walks.