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:

Accepted:

Published online:

DOI: 10.5802/alco.60

Keywords: representation stability, subspace arrangement, symmetric functions

Rapp, Artur ^{1}

@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}, zbl = {1427.55012}, mrnumber = {3997513}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.60/} }

TY - JOUR AU - Rapp, Artur TI - Representation stability on the cohomology of complements of subspace arrangements JO - Algebraic Combinatorics PY - 2019 SP - 603 EP - 611 VL - 2 IS - 4 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.60/ DO - 10.5802/alco.60 LA - en ID - ALCO_2019__2_4_603_0 ER -

%0 Journal Article %A Rapp, Artur %T Representation stability on the cohomology of complements of subspace arrangements %J Algebraic Combinatorics %D 2019 %P 603-611 %V 2 %N 4 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.60/ %R 10.5802/alco.60 %G en %F 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/articles/10.5802/alco.60/

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