Free group representations from vector-valued multiplicative functions, II

Let (Formula presented.) be a non-commutative free group on finitely many generators. In a previous work two of the authors have constructed the class of multiplicative representations of (Formula presented.) and proved them irreducible as representation of (Formula presented.). In this paper we analyze multiplicative representations as representations of (Formula presented.) and we prove a criterium for irreducibility based on the growth of their matrix coefficients.