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

被引：40

作者：

Church, Thomas ^{[1
]}

Ellenberg, Jordan S. ^{[2
]}

Farb, Benson ^{[3
]}

机构：

[1] Stanford Univ, Dept Math, Stanford, CA 94305 USA

[2] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

[3] Univ Chicago, Dept Math, Chicago, IL 60637 USA

来源：

ALGEBRAIC TOPOLOGY: APPLICATIONS AND NEW DIRECTIONS | 2014年 / 620卷

基金：

美国国家科学基金会;

关键词：

POINTS;

D O I：

10.1090/conm/620/12395

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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 (resp. maximal tori in GL(n)(F-q)) with specified properties related to V. In particular, we explain how representation stability in cohomology, in the sense of [Church, Farb, 2013] and [Church, Ellenberg, Farb, 2012] corresponds to asymptotic stability of various point counts as n -> infinity.

引用

页码：1 / 54

页数：54

共 26 条

[1]

Arnold V.I., 1970, T MOSK MATEM OBVA, V21, P27

[2] ON RANDOM POLYNOMIALS OVER FINITE-FIELDS [J].