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 = {https://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/

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