Numerical groups

TL;DR

This paper characterizes numerical groups of matrices over number fields as those whose elements have algebraic integer traces, providing a criterion for when such groups are numerical.

Contribution

It establishes a necessary and sufficient condition for irreducible or completely reducible subgroups of GL(n,K) to be numerical based on trace properties.

Findings

Numerical groups are characterized by algebraic integer traces.

Irreducible or completely reducible subgroups are numerical iff traces are algebraic integers.

Provides examples illustrating the characterization.

Abstract

A group of matrices $G$ with entries in a number field $K$ is defined to be numerical if $G$ has a finite index subgroup of matrices whose entries are algebraic integers. It is shown that an irreducible or completely reducible subgroup of $GL(n,K)\subset GL(n,\mathbb{C})$ is numerical if and only if the traces of its elements are algebraic integers. Some examples are given.