On characterization of monomial representations of discrete supersolvable groups

TL;DR

This paper characterizes monomial representations of discrete supersolvable groups, showing they are equivalent to having finite weight, and explores related properties including examples with the infinite dihedral group.

Contribution

It establishes a characterization of monomial representations in terms of finite weight and extends Schur's lemma to certain induced representations.

Findings

Monomial representations of discrete supersolvable groups are exactly those with finite weight.

Schur's lemma converse holds for certain induced representations of finitely generated groups.

Infinite dihedral group is demonstrated to be a monomial group.

Abstract

We prove that an abstract (possibly infinite dimensional) complex irreducible representation of a discrete supersolvable group is monomial if and only if it has finite weight. We also prove a general result that implies converse of Schur's lemma holds true for certain induced representations of finitely generated discrete groups. At last, we work out example of infinite dihedral group and prove that it is a monomial group.