Representation stability in cohomology and asymptotics for families of varieties over finite fields

Abstract: We consider two families X_n of varieties on which the symmetric group S_n acts: the configuration space of n points in C and the space of n linearly independent lines in C^n. Given an irreducible S_n-representation V, one can ask how the multiplicity of V in the cohomology groups H*(X_n;Q) varies with n. We explain how the Grothendieck-Lefschetz Fixed Point Theorem converts a formula for this multiplicity to a formula for the number of polynomials over F_q (or maximal tori in GL_n(F_q), respectively) with specified properties related to V. In particular, we explain how representation stability in cohomology, in the sense of [CF, arXiv:1008.1368] and [CEF, arXiv:1204.4533], corresponds to asymptotic stability of various point counts as n goes to infinity.