Multiplicative functions on free groups and irreducible representations

M. Gabriella Kuhn and Tim Steger

Vol. 169 (1995), No. 2, 311–334

DOI: 10.2140/pjm.1995.169.311

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| + |y|. We characterize those positive definite multiplicative functions for which the associated representation of Γ is irreducible.

Mathematical Subject Classification 2000

Primary: 20E05

Secondary: 22D10, 43A35

Milestones

Received: 20 October 1992

Revised: 10 January 1994

Published: 1 June 1995