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.

57 pages

References in corpus (3)

• FI-modules and stability for representations of symmetric groups
• Discriminants in the Grothendieck Ring
• Some stable homology calculations and Occam's razor for Hodge structures

Cited by in corpus (19)

• FI-modules and stability for representations of symmetric groups
• Discriminants in the Grothendieck Ring
• FI-modules and the cohomology of modular representations of symmetric groups
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• Arithmetic functions in short intervals and the symmetric group
• The geometric average size of Selmer groups over function fields
• Homology of braid groups, the Burau representation, and ${\mathbb{F}}_q$-points on superelliptic curves
• Splitting Behavior of $S_n$-Polynomials
• Higher moments of arithmetic functions in short intervals: a geometric perspective
• Polynomial splitting measures and cohomology of the pure braid group
• Motivic random variables and representation stability I: Configuration spaces
• A family of measures on symmetric groups and the field with one element
• Analytic number theory for 0-cycles
• Configurations of noncollinear points in the projective plane
• Dimension-independent statistics of $G{l}_n(F_q)$ via character polynomials
• Factorization statistics and bug-eyed configuration spaces
• Stability in the cohomology of the space of complex irreducible polynomials in several variables
• First-degree prime ideals of composite extensions
• A geometric approach to counting norms in cyclic extensions of function fields