Burnside type theorems in real and quaternion settings

TL;DR

This paper extends Burnside's theorems to real and quaternionic matrix settings, establishing conditions under which irreducible semigroups generate the entire matrix algebra and characterizing their structure.

Contribution

It proves Burnside type theorems for irreducible semigroups of matrices over reals and quaternions, revealing their algebraic structure and uniqueness properties.

Findings

Irreducible semigroups of triangularizable matrices in $M_n(\mathbb{R})$ span the entire matrix algebra.

$M_n(\mathbb{R})$ is the unique irreducible subalgebra spanned by such semigroups.

Up to similarity, $M_n(\mathbb{R})$ is the only irreducible $\mathbb{R}$-algebra in $M_n(\mathbb{H})$ with real spectra.

Abstract

In this note, we consider irreducible semigroups of real, complex, and quaternionic matrices with real spectra. We prove Burnside type theorems in the settings of reals and quaternions. First, we prove that an irreducible semigroup of triangularizable matrices in $M_n(\mathbb{R})$ contains a vector space basis for $M_n(\mathbb{R})$. In other words, $M_n(\mathbb{R})$ is the only irreducible subalgebra of itself that is spanned by an irreducible semigroup of triangularizable matrices in $M_n(\mathbb{R})$. Next, we use this result to show that, up to similarity, $M_n(\mathbb{R})$ is the only irreducible $\mathbb{R}$-algebra in $M_n(\mathbb{H})$ that is spanned by an irreducible semigroup of matrices in $M_n(\mathbb{H})$ with real spectra. Some consequences of our mains results are presented.