Proportion of cyclic matrices in maximal reducible matrix algebras

Abstract

Let $\mathrm{M}(V)=\mathrm{M}(n,{\mathbb{F}}_q)$ denote the algebra of $n\times n$ matrices over ${\mathbb{F}}_q$, and let $\mathrm{M}(V{)}_U$ 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 $\mathrm{M}(V{)}_U$ is at least $q^{-2}\left(1+c_1{q}^{-1}\right)$, and at most $q^{-2}\left(1+c_2{q}^{-1}\right)$, where $c_1$ and $c_2$ are constants independent of $n,r$, and $q$. The constants $c_1=-\frac{4}{3}$ and $c_2=\frac{35}{3}$ suffice.

11 pages

References in corpus (3)

• Planar open books, monodromy factorizations, and symplectic fillings
• Minimality and irreducibility of symplectic four-manifolds
• Monodromy Substitutions and Rational Blowdowns