Groups with faithful irreducible projective unitary representations

Bachir Bekka (1) , Pierre de La Harpe (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Section de mathématiques [Genève]

For a countable group $\Gamma$ and a multiplier $\zeta \ Z^2(\Gamma,T)$, we study the property of $\Gamma$ having a unitary projective -representation which is both irreducible and projectively faithful. Theorem 1 shows that this property is equivalent to $\Gamma$ being the quotient of an appropriate group by its centre. Theorem 4 gives a criterion in terms of the minisocle of $\Gamma$. Several examples are described to show the existence of various behaviours.

Keywords: Projective unitary representations; Irreducible representations; Projectively faithful representations; Central extensions; Stem covers; Capable groups

Domains: Representation Theory [math.RT]

Bachir Bekka, Pierre de La Harpe. Groups with faithful irreducible projective unitary representations. Forum Mathematicum, 2013, 25 (3), pp.505-531. ⟨10.1515/FORM.2011.126⟩. ⟨hal-00564624⟩