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

Thomas M. Church, Jordan S. Ellenberg, Benson Farb

Type: Other

Publication Date: 2014-01-01

Citations: 43

DOI: https://doi.org/10.1090/conm/620/12395

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 (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 [CF] and [CEF], corresponds to asymptotic stability of various point counts as n → ∞.

Locations

Contemporary mathematics - American Mathematical Society - View
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Works Cited by This (28)

Action	Title	Year	Authors
+	Free Lie Algebras	1993	Christophe Reutenauer
+	The problem of coincidences	1980	Lajos Tak�cs
+	Linear algebraic groups	1966	Armand Borel
+	Young Tableaux: With Applications to Representation Theory and Geometry	1996	William Fulton
+	Representations of Finite Chevalley Groups	1978	G. Lusztig
+	Finite Groups of Lie Type: Conjugacy Classes and Complex Characters	1993	Roger W. Carter
+	On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes	1986	G Lehrer Louis Solomon
+	Equivariant Poincaré Polynomials and Counting Points over Finite Fields	2002	Mark Kisin G Lehrer
+	Rational tori, semisimple orbits and the topology of hyperplane complements	1992	G Lehrer
+	Rational points and cohomology of discriminant varieties	2003	G Lehrer