We study representation stability in the sense of Church and Farb of sequences of cohomology groups of complements of arrangements of linear subspaces in real and complex space as ${S}_{n}$-modules. We consider arrangements of linear subspaces defined by sets of diagonal equalities ${x}_{i}={x}_{j}$ and invariant under the action of ${S}_{n}$ which permutes the coordinates. We provide bounds for the point when stabilization occurs and an alternative proof of the fact that stabilization happens. The latter is a special case of very general stabilization results proved independently by Gadish and by Petersen; for the pure braid space the result is part of the work of Church and Farb. For the latter space, better stabilization bounds were obtained by Hersh and Reiner.

Revised: 2018-06-15

Accepted: 2019-01-16

Published online: 2019-08-01

DOI: https://doi.org/10.5802/alco.60

Classification: 55-XX, 05E10

Keywords: representation stability, subspace arrangement, symmetric functions

@article{ALCO_2019__2_4_603_0, author = {Rapp, Artur}, title = {Representation stability on the cohomology of complements of subspace arrangements}, journal = {Algebraic Combinatorics}, pages = {603--611}, publisher = {MathOA foundation}, volume = {2}, number = {4}, year = {2019}, doi = {10.5802/alco.60}, mrnumber = {3997513}, zbl = {1427.55012}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2019__2_4_603_0/} }

Rapp, Artur. Representation stability on the cohomology of complements of subspace arrangements. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 603-611. doi : 10.5802/alco.60. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_603_0/

[1] Homological stability for configuration spaces of manifolds, Invent. Math., Volume 188 (2012) no. 2, pp. 465-504 | Article | MR 2909770 | Zbl 1244.55012

[2] Representation theory and homological stability, Adv. Math., Volume 245 (2013), pp. 250-314 | Article | MR 3084430 | Zbl 1300.20051

[3] Representation stability for families of linear subspace arrangements, Adv. Math., Volume 322 (2017), pp. 341-377 | Article | MR 3720801 | Zbl 1377.14012

[4] Representation stability for cohomology of configuration spaces in ${\mathbb{R}}^{d}$, Int. Math. Res. Not. IMRN, Volume 2017 (2017) no. 5, pp. 1433-1486 (With an appendix written jointly with Steven Sam) | Article | MR 3658170 | Zbl 1404.20009

[5] Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications) | MR 1354144 | Zbl 0824.05059

[6] A spectral sequence for stratified spaces and configuration spaces of points, Geom. Topol., Volume 21 (2017) no. 4, pp. 2527-2555 | Article | MR 3654116 | Zbl 06726529

[7] The homology representations of the $k$-equal partition lattice, Trans. Amer. Math. Soc., Volume 349 (1997) no. 3, pp. 935-954 | Article | MR 1389790 | Zbl 0863.05082

[8] Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc., Volume 349 (1997) no. 4, pp. 1389-1420 | Article | MR 1340186 | Zbl 0945.05067