FI-modules: a new approach to stability for Sn-representations

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of n distinct ordered points on an arbitrary manifold; • the diagonal coinvariant algebra on r sets of n variables; • the cohomology and tautological ring of the moduli space of n-pointed curves; • the space of polynomials on rank varieties of n× n matrices; • the subalgebra of the cohomology of the genus n Torelli group generated by H; and more. The symmetric group Sn acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. FI-modules are a refinement of Church–Farb’s theory of representation stability for representations of Sn. In this framework, a complicated sequence of Sn-representations becomes a single FI-module, and representation stability becomes finite generation. FI-modules also shed light on classical results. From this point of view, Murnaghan’s theorem on the stability of Kronecker coefficients is not merely an assertion about a list of numbers, but becomes a structural statement about a single mathematical object.