Perron matrix semigroups

TL;DR

This paper investigates conditions under which multiplicative semigroups of real matrices share a common invariant cone, providing classifications for low dimensions and characterizing exceptions in higher dimensions.

Contribution

It proves that irreducible Perron semigroups have a common invariant cone under mild assumptions and classifies all such semigroups for dimensions up to four.

Findings

Irreducible Perron semigroups possess a common invariant cone under certain conditions.

Complete classification of Perron semigroups for dimensions up to four.

Identification of classes of Perron semigroups without invariant cones in higher dimensions.

Abstract

We consider multiplicative semigroups of real dxd matrices. A semigroup S is called Perron if each of its matrices has a Perron eigenvalue, i.e., an eigenvalue equal to the spectral radius. If all matrices of S leave a proper convex cone invariant, then S is Perron. Our main result asserts the converse: every irreducible Perron semigroup possesses a common invariant cone, provided that some mild assumptions are satisfied. This gives conditions for a set of matrices to share a common invariant cone, which is an important property widely studied in the literature. Then we address the problem to characterize the exceptions, when a Perron semigroup does not have an invariant cone. For d\le 4, all Perron semigroups are classified. For higher dimensions~$d$, several classes of such semigroups are found.