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Representation stability in cohomology and asymptotics for families of varieties over finite fields. (English) Zbl 1388.14148
Tillmann, Ulrike (ed.) et al., Algebraic topology: applications and new directions. Stanford symposium on algebraic topology: applications and new directions, Stanford University, Stanford, CA, USA, July 23–27, 2012. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9474-3/pbk; 978-1-4704-1855-7/ebook). Contemporary Mathematics 620, 1-54 (2014).
Summary: We consider two families \(X_n\) of varieties on which the symmetric group \(S_n\) acts: the configuration space of \(n\) points in \(\mathbb C\) and the space of \(n\) linearly independent lines in \(\mathbb C^n\). 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 \(\mathbb F_q\) (or maximal tori in \(\mathrm{GL}_n(\mathbb F_q)\), respectively) with specified properties related to \(V\). In particular, we explain how representation stability in cohomology, in the sense of T. Church and B. Farb [Adv. Math. 245, 250–314 (2013; Zbl 1300.20051)] and with T. Ellenberg [Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)], corresponds to asymptotic stability of various point counts as \(n\) goes to infinity.
For the entire collection see [Zbl 1294.00030].

MSC:
14N20 Configurations and arrangements of linear subspaces
11T06 Polynomials over finite fields
14F99 (Co)homology theory in algebraic geometry
14G15 Finite ground fields in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
20C30 Representations of finite symmetric groups
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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