Representation stability on the cohomology of complements of subspace arrangements
Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 603-611.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/alco.60
Classification : 55-XX, 05E10
Mots clés : representation stability, subspace arrangement, symmetric functions
Rapp, Artur 1

1 Philipps-Universität Marburg Fachbereich Mathematik und Informatik Hans-Meerweinstr. 6 35032 Marburg, Germany
@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 = {http://www.numdam.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  - http://www.numdam.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 http://www.numdam.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, Tome 2 (2019) no. 4, pp. 603-611. doi : 10.5802/alco.60. http://www.numdam.org/articles/10.5802/alco.60/

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

[2] Church, Thomas; Farb, Benson Representation theory and homological stability, Adv. Math., Volume 245 (2013), pp. 250-314 | DOI | MR | Zbl

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

[4] Hersh, Patricia; Reiner, Victor Representation stability for cohomology of configuration spaces in d , Int. Math. Res. Not. IMRN, Volume 2017 (2017) no. 5, pp. 1433-1486 (With an appendix written jointly with Steven Sam) | DOI | MR | Zbl

[5] Macdonald, Ian Grant 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 | Zbl

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

[7] Sundaram, Sheila; Wachs, Michelle The homology representations of the k-equal partition lattice, Trans. Amer. Math. Soc., Volume 349 (1997) no. 3, pp. 935-954 | DOI | MR | Zbl

[8] Sundaram, Sheila; Welker, Volkmar Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc., Volume 349 (1997) no. 4, pp. 1389-1420 | DOI | MR | Zbl

Cité par Sources :