Irreducible representations of finitely generated nilpotent groups

TL;DR

This paper characterizes when irreducible complex representations of finitely generated nilpotent groups are monomial, linking this property to finite weight, and explores conditions for irreducibility in induced representations over arbitrary fields.

Contribution

It proves the conjecture by Parshin relating monomiality and finite weight for these groups and extends results on irreducibility criteria to a broader algebraic context.

Findings

Irreducible representations are monomial iff they have finite weight.

Irreducibility of certain induced representations follows from Schur irreducibility.

Results hold over arbitrary fields, not just complex numbers.

Abstract

We prove that irreducible complex representations of finitely generated nilpotent groups are monomial if and only if they have finite weight, which was conjectured by Parshin. Note that we consider (possibly, infinite-dimensional) representations without any topological structure. Besides, we prove that for certain induced representations, irreducibility is implied by Schur irreducibility. Both results are obtained in a more general form for representations over an arbitrary field.