Proportion of cyclic matrices in maximal reducible matrix algebras

TL;DR

This paper investigates the proportion of cyclic matrices within maximal reducible matrix algebras over finite fields, providing bounds that are independent of matrix size and subspace dimension.

Contribution

It establishes explicit bounds on the density of non-cyclic matrices in maximal reducible subalgebras, advancing understanding of their structure over finite fields.

Findings

Density of non-cyclic matrices is at least q^{-2}(1 - 4/3 q^{-1})

Density of non-cyclic matrices is at most q^{-2}(1 + 35/3 q^{-1})

Constants are independent of n, r, and q

Abstract

Let ${\rm M}(V)={\rm M}(n,\mathbb{F}_q)$ denote the algebra of $n\times n$ matrices over $\mathbb{F}_q$, and let ${\rm M}(V)_U$ denote the (maximal reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of $V=\mathbb{F}_q^n$ where $0<r<n$. We prove that the density of non-cyclic matrices in ${\rm M}(V)_U$ is at least $q^{-2}\left(1+c_1q^{-1}\right)$, and at most $q^{-2}\left(1+c_2q^{-1}\right)$, where $c_1$ and $c_2$ are constants independent of $n,r$, and $q$. The constants $c_1=-\frac43$ and $c_2=\frac{35}3$ suffice.