FIW-modules and stability criteria for representations of classical Weyl groups

In this paper we develop machinery for studying sequences of representations of any of the three families of classical Weyl groups, extending work of Church, Ellenberg, Farb, and Nagpal [7,8] on the symmetric groups S n to the signed permutation groups B n and the even-signed permutation groups D n . For each family W n , we present an algebraic framework where a sequence V n of W n -representations is encoded into a single object we call an FI W -module . We prove that if an FI W -module V satisfies a simple finite generation condition then the structure of the sequence is highly constrained. One consequence is that the sequence is uniformly representation stable in the sense of Church–Farb, that is, the pattern of irreducible representations in the decomposition of each V n eventually stabilizes in a precise sense. Using the theory developed here we obtain new results about the cohomology of generalized flag varieties associated to the classical Weyl groups, and more generally the r -diagonal coinvariant algebras. We analyze the algebraic structure of the category of FI W -modules, and introduce restriction and induction operations that enable us to study interactions between the three families of groups. We use this theory to prove analogues of Murnaghan's 1938 stability theorem for Kronecker coefficients for the families B n and D n . The theory of FI W -modules gives a conceptual framework for stability results such as these.