Representation zeta functions of wreath products with finite groups

الأصل · EN

Let G be a group which has for all n a finite number rₙ(G) of irreducible complex linear representations of dimension n. Let ζ(G,s) = ∑ₙ₌₁∞ rₙ(G) n⁻ˢ be its representation zeta function. First, in case G is a permutational wreath product of H with a permutation group Q acting on a finite set X, we establish a formula for ζ(G,s) in terms of the zeta functions of H and of subgroups of Q, and of the Moebius function associated with the lattice of partitions of X in orbits under subgroups of Q. Then, we consider groups W(Q,k) which are k-fold iterated wreath products of Q, and several related infinite groups W(Q), including the profinite group, a locally finite group, and several finitely generated groups, which are all isomorphic to a wreath product of themselves with Q. Under convenient hypotheses (in particular Q should be perfect), we show that rₙ(W(Q)) is finite for all n, and we establish that the Dirichlet series ζ(W(Q),s) has a finite and positive abscissa of convergence s₀. Moreover, the function ζ(W(Q),s) satisfies a remarkable functional equation involving ζ(W(Q),es) for e=1,...,|X|. As a consequence of this, we exhibit some properties of the function, in particular that ζ(W(Q),s) has a singularity at s₀, a finite value at s₀, and a Puiseux expansion around s₀. We finally report some numerical computations for Q the simple groups of order 60 and 168.