Abstract
From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we show that the rational cohomology stabilizes as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with FI\(_W\)-module theory which Wilson developed and used in the linear case. A key to the proof relies on a combinatorial description, using labelled partitions, of the poset of connected components of intersections of subvarieties in the arrangement.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnol’d, V.I.: The cohomology ring of the colored braid group. Math. Notes 5(2), 138–140 (1969)
Barcelo, H., Ihrig, E.: Lattices of parabolic subgroups in connection with hyperplane arrangements. J. Algebraic Combin 9(1), 5–24 (1999). doi:10.1023/A:118607830027
Bezrukavnikov, R.: Koszul DG-algebras arising from configuration spaces. Geom. Funct. Anal. 4(2), 119–135 (1994)
Bibby, C.: Cohomology of abelian arrangements. Proc. Am. Math. Soc. 144(7), 3093–3104 (2016)
Brieskorn, E.: Sur les groupes de tresses [d’aprés V. I. Arnol’d]. Séminaire Bourbaki, 24éme année (1971/1972), Exp. No. 401, Springer-Verlag, Berlin, 1973, pp. 21–44. Lecture Notes in Math., Vol. 317
Chen, W.: Twisted cohomology of configuration spaces of maximal tori via point-counting (2016). arXiv:1603.03931
Church, T.: Homological stability for configuration spaces of manifolds. Invent. Math. 188(2), 465–504 (2012)
Church, T., Ellenberg, J.S., Farb, B.: FI-modules and stability for representations of symmetric groups. Duke Math. J. 164(9), 1833–1910 (2015)
Church, T., Farb, B.: Representation theory and homological stability. Adv. Math. 245, 250–314 (2013)
De Concini, C., Procesi, C.: On the geometry of toric arrangements. Transform. Groups 10(3–4), 387–422 (2005)
Hersh, P., Reiner, V.: Representation stability for the cohomology of configuration spaces in \(\mathbb{R}^d\). Int. Math. Res. Notices (2016). doi:10.1093/imrn/rnw060
Jiménez Rolland, R.: On the cohomology of pure mapping class groups as FI-modules. J. Homotopy Relat. Struct. 10(3), 401–424 (2015)
Kupers, A., Miller, J.: Representation stability for homotopy groups of configuration spaces. J. Reine Angew. Math. (2015). doi:10.1515/crelle-2015-0038
Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math. 56(2), 167–189 (1980)
Totaro, B.: Configuration spaces of algebraic varieties. Topology 35(4), 1057–1067 (1996)
Wilson, J.C.H.: \({\rm FI}_{\cal{W}}\)-modules and stability criteria for representations of classical Weyl groups. J. Algebra 420, 269–332 (2014)
Wilson, J.C.H.: \({\rm FI}_{\cal{W}}\)-modules and constraints on classical Weyl group characters. Math. Z. 281(1–2), 1–42 (2015)
Acknowledgements
The author wishes to thank Nick Proudfoot and Benson Farb for piquing her interest in this project, and Graham Denham for helpful conversations. Many thanks to Jenny Wilson for pointing out and helping to fix an error in an earlier draft. Finally, the author is grateful to the referees for many useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bibby, C. Representation stability for the cohomology of arrangements associated to root systems. J Algebr Comb 48, 51–75 (2018). https://doi.org/10.1007/s10801-017-0792-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-017-0792-0