$$\text {FI}_{\mathcal {W}}$$ FI W -modules and constraints on classical Weyl group characters
In this paper we study the characters of sequences of representations of any of the three families of classical Weyl groups $$\mathcal {W}_n$$
 : the symmetric groups, the signed permutation groups (hyperoctahedral groups), or the even-signed permutation groups. Our results extend work of Church et al. (Duke Math J, 2012. arXiv:1204.4533
 ; Geom Topol 18(5):2951–2984, 2014) on the symmetric groups. We use the concept of an $$\text {FI}_{\mathcal {W}}$$
 -module, an algebraic object that encodes the data of a sequence of $$\mathcal {W}_n$$
 -representations with maps between them, defined in the author’s recent work (J Algebra 420:269–332, 2014). We show that if a sequence $$\{V_n\}$$
 of $$\mathcal {W}_n$$
 -representations has the structure of a finitely generated $$\text {FI}_{\mathcal {W}}$$
 -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 $$\text {FI}_{\mathcal {W}}$$
 -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).