Multiplicative functions on free groups and irreducible representations

Abstract

Let Γ be a free group on infinitely many generators. Fix a basis for Γ and for any group element x, denote by |x| its length with respect to this basis. Let e denote the group identity. A multiplicative function φ on Γ is a function satisfying the conditions φ(e) = 1 and φ(xy) = φ{x)φ(y) whenever \xy = \x -f \y\. We characterize those positive definite multiplicative functions for which the associated representation of Γ is irreducible.