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
1
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