Groups with faithful irreducible projective unitary representations

TL;DR

This paper investigates when countable groups admit irreducible, faithful projective unitary representations, establishing a characterization related to quotients by the group's center and providing criteria involving the minisocle.

Contribution

It provides a new characterization of groups with faithful irreducible projective representations, linking this property to quotients by the center and criteria involving the minisocle.

Findings

Equivalence between the existence of such representations and being a quotient by the center.

A criterion based on the minisocle of the group.

Examples illustrating different behaviors of these groups.

Abstract

For a countable group G and a multiplier c on G with values in the circle, we study the property of G having a unitary projective c-representation which is both irreducible and projectively faithful. We show that this property is equivalent to G being the quotient of an appropriate group by its centre. A criterion is given in terms of the minisocle of G. Several examples are described to show the existence of various behaviours.