Contemporary Mathematics

Volume 620, 2014

http://dx.doi.org/10.1090/conm/620/12395

Representation stability in cohomology and

asymptotics for families of varieties over finite fields

Thomas Church, Jordan S. Ellenberg, and Benson Farb

Abstract. We consider two families Xn of varieties on which the symmetric

group Sn acts: the configuration space of n points in C and the space of n

linearly independent lines in

Cn.

Given an irreducible Sn-representation V ,

one can ask how the multiplicity of V in the cohomology groups

H∗(Xn;

Q)

varies with n. We explain how the Grothendieck–Lefschetz Fixed Point The-

orem converts a formula for this multiplicity to a formula for the number of

polynomials over Fq

(resp. maximal tori in GLn(Fq)) with specified properties

related to V . In particular, we explain how representation stability in coho-

mology, in the sense of [Church, Farb, 2013] and [Church, Ellenberg,

Farb, 2012] corresponds to asymptotic stability of various point counts as

n → ∞.

1. Introduction

In this paper we consider certain families X1,X2,... of algebraic varieties for

which Xn is endowed with a natural action of the permutation group Sn. In par-

ticular Sn acts on the complex solution set Xn(C), and so each cohomology group

Hi(Xn(C))

has the structure of an Sn-representation. We will attach to Xn a

variety Yn over the finite field Fq. The goal of this paper is to explain how rep-

resentation stability for

Hi(Xn(C)),

in the sense of [CF] and [CEF], corresponds

to asymptotic stability for certain counting problems on the Fq-points Yn(Fq), and

vice versa.

We will concentrate on two such families of varieties in this paper. The first

family is the configuration space of n distinct points in C:

Xn(C) = PConfn(C) = (z1,...,zn) zi ∈ C,zi = zj

In this case Yn(Fq) is the space Confn(Fq) of monic squarefree degree-n polynomials

in Fq[T ]. The second family is the space of n linearly independent lines in Cn:

Xn(C) = (L1,...,Ln) Li a line in

Cn,

L1,...,Ln linearly independent

In this case Yn(Fq) is the space parametrizing the set of maximal tori in the finite

group GLn(Fq). In both cases, the action of Sn on Xn(C) simply permutes the

points zi or the lines Li.

The relation between Xn(C) and Yn(Fq) is given by the Grothendieck–Lefschetz

fixed point theorem in ´ etale cohomology. For any irreducible Sn-representation Vn

with character χn, the Grothendieck–Lefschetz theorem with twisted coeﬃcients Vn

2010 Mathematics Subject Classification. Primary 11T06, 14F20, 55N99.

c 2014 American Mathematical Society

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