Irreducible representations of certain nilpotent groups of finite rank

TL;DR

This paper investigates the structure of irreducible representations of certain nilpotent groups of finite rank, establishing conditions under which these representations can be induced from primitive representations of subgroups.

Contribution

It provides a new characterization of irreducible representations of torsion-free minimax nilpotent groups of class 2, linking them to induced representations from subgroups under specific field conditions.

Findings

Existence of a subgroup and primitive representation inducing the given irreducible representation.

The quotient of the subgroup by the kernel of the primitive representation is finitely generated.

Conditions on the characteristic of the field relative to the spectrum of the group.

Abstract

In the paper we study irreducible representations of some nilpotent groups of finite abelian total rank. The main result of the paper states that if a torsion-free minimax group $G$ of nilpotency class 2 admits a faithful irreducible representation $\varphi $ over a finitely generated field $k$ such that $chark \notin Sp(G)$ then there exist a subgroup $N$ and an irreducible primitive representation $\psi $ of the subgroup $N$ over $k$ such that the representation $\varphi $ is induced from $\psi $ and the quotient group $N/Ker\psi $ is finitely generated.