On monomial representations of finitely generated nilpotent groups

TL;DR

This paper provides a shorter proof of a conjecture characterizing monomial irreducible representations of finitely generated nilpotent groups and describes finite dimensional representations of specific two-step nilpotent groups.

Contribution

It offers a concise proof of Parshin's conjecture and characterizes finite dimensional irreducible representations of two-step nilpotent groups.

Findings

Shorter proof of Parshin's conjecture on monomial representations

Characterization of finite dimensional irreducible representations of two-step nilpotent groups

Full description of representations of two-step groups with rank-one center

Abstract

A result of D. Segal states that every complex irreducible representation of a finitely generated nilpotent group $G$ is monomial if and only if $G$ is abelian-by-finite. A conjecture of A. N. Parshin, recently proved affirmatively by I.V. Beloshapka and S. O. Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture combining the ideas of I. D. Brown and P. C. Kutzko. We also characterize finite dimensional irreducible representations of two step nilpotent groups and also provide a full description of the finite dimensional representations of two step groups whose center has rank one.