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\(\text {FI}_{\mathcal {W}}\)-modules and constraints on classical Weyl group characters

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Abstract

In this paper we study the characters of sequences of representations of any of the three families of classical Weyl groups \(\mathcal {W}_n\): the symmetric groups, the signed permutation groups (hyperoctahedral groups), or the even-signed permutation groups. Our results extend work of Church et al. (Duke Math J, 2012. arXiv:1204.4533; Geom Topol 18(5):2951–2984, 2014) on the symmetric groups. We use the concept of an \(\text {FI}_{\mathcal {W}}\)-module, an algebraic object that encodes the data of a sequence of \(\mathcal {W}_n\)-representations with maps between them, defined in the author’s recent work (J Algebra 420:269–332, 2014). We show that if a sequence \(\{V_n\}\) of \(\mathcal {W}_n\)-representations has the structure of a finitely generated \(\text {FI}_{\mathcal {W}}\)-module, then there are substantial constraints on the growth of the sequence and the structure of the characters: for \(n\) large, the dimension of \(V_n\) is equal to a polynomial in \(n\), and the characters of \(V_n\) are given by a character polynomial in signed-cycle-counting class functions, independent of \(n\). We determine bounds the degrees of these polynomials. We continue to develop the theory of \(\text {FI}_{\mathcal {W}}\)-modules, and we apply this theory to obtain new results about a number of sequences associated to the classical Weyl groups: the cohomology of complements of classical Coxeter hyperplane arrangements, and the cohomology of the pure string motion groups (the groups of symmetric automorphisms of the free group).

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Acknowledgments

I am grateful to Benson Farb, Tom Church, Jordan Ellenberg, Rita Jimenez Rolland, and Peter May for numerous helpful discussions about this project. Thanks are especially due to Benson Farb, my PhD advisor, for extensive guidance and feedback throughout the project. This work was supported in part by a PGS D Scholarship from the Natural Sciences and Engineering Research Council of Canada.

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  1. Stanford University, Stanford, CA, USA

    Jennifer C. H. Wilson

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Wilson, J.C.H. \(\text {FI}_{\mathcal {W}}\)-modules and constraints on classical Weyl group characters. Math. Z. 281, 1–42 (2015). https://doi.org/10.1007/s00209-015-1473-0

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