FI_W-modules and constraints on classical Weyl group characters

Abstract

In this paper we study the characters of sequences of representations of any of the three families of classical Weyl groups W_n: the symmetric groups, the signed permutation groups (hyperoctahedral groups), or the even-signed permutation groups. Our results extend work of Church, Ellenberg, Farb, and Nagpal on the symmetric groups. We use the concept of an FI_W-module, an algebraic object that encodes the data of a sequence of W_n-representations with maps between them, defined in the author's recent work ArXiv:1309.3817. We show that if a sequence {V_n} of W_n-representations has the structure of a finitely generated FI_W-module, then there are substantial constraints on the growth of the sequence and the structure of the characters: for n large, the dimension of V_n is equal to a polynomial in n, and the characters of V_n are given by a character polynomial in signed-cycle-counting class functions, independent of n. We determine bounds the degrees of these polynomials. We continue to develop the theory of FI_W-modules, and we apply this theory to obtain new results about a number of sequences associated to the classical Weyl groups: the cohomology of complements of classical Coxeter hyperplane arrangements, and the cohomology of the pure string motion groups (the groups of symmetric automorphisms of the free group).