We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/alco.76
Mots-clés : Stanley–Stembridge conjecture, symmetric functions, e-positivity, Hessenberg varieties, abelian ideal
@article{ALCO_2019__2_6_1059_0, author = {Harada, Megumi and Precup, Martha E.}, title = {The cohomology of abelian {Hessenberg} varieties and the {Stanley{\textendash}Stembridge} conjecture}, journal = {Algebraic Combinatorics}, pages = {1059--1108}, publisher = {MathOA foundation}, volume = {2}, number = {6}, year = {2019}, doi = {10.5802/alco.76}, zbl = {07140425}, mrnumber = {4049838}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.76/} }
TY - JOUR AU - Harada, Megumi AU - Precup, Martha E. TI - The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture JO - Algebraic Combinatorics PY - 2019 SP - 1059 EP - 1108 VL - 2 IS - 6 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.76/ DO - 10.5802/alco.76 LA - en ID - ALCO_2019__2_6_1059_0 ER -
%0 Journal Article %A Harada, Megumi %A Precup, Martha E. %T The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture %J Algebraic Combinatorics %D 2019 %P 1059-1108 %V 2 %N 6 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.76/ %R 10.5802/alco.76 %G en %F ALCO_2019__2_6_1059_0
Harada, Megumi; Precup, Martha E. The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture. Algebraic Combinatorics, Tome 2 (2019) no. 6, pp. 1059-1108. doi : 10.5802/alco.76. http://www.numdam.org/articles/10.5802/alco.76/
[1] The cohomology rings of regular nilpotent Hessenberg Varieties in Lie type A, Int. Math. Res. Not., Volume 2019 (2019) no. 17, pp. 5316-5388 | DOI | MR
[2] The cohomology rings of regular semisimple Hessenberg varieties for
[3] Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties, Adv. Math., Volume 329 (2018), pp. 955-1001 | DOI | MR | Zbl
[4] Representation theory of the symmetric groups, Cambridge Studies in Advanced Mathematics, 121, Cambridge University Press, Cambridge, 2010, xvi+412 pages | DOI | MR | Zbl
[5] On
[6] Hessenberg varieties, Trans. Amer. Math. Soc., Volume 332 (1992) no. 2, pp. 529-534 | DOI | MR | Zbl
[7] On the topology of the Hessenberg varieties of a matrix, Ph. D. Thesis, Washington University in St. Louis (USA) (1987) | MR
[8] Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages | MR | Zbl
[9] Incomparability graphs of
[10] A chromatic symmetric function in noncommuting variables, J. Algebraic Combin., Volume 13 (2001) no. 3, pp. 227-255 | DOI | MR | Zbl
[11] A modular relation for the chromatic symmetric functions of (3+1)-free posets (2013) (https://arxiv.org/abs/1306.2400)
[12] A second proof of the Shareshian–Wachs conjecture, by way of a new Hopf algebra (2016) (https://arxiv.org/abs/1601.05498)
[13] The
[14] Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1972, xii+169 pages | DOI | MR | Zbl
[15] Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2), Volume 74 (1961), pp. 329-387 | DOI | MR | Zbl
[16] The set of abelian ideals of a Borel subalgebra, Cartan decompositions, and discrete series representations, Int. Math. Res. Not., Volume 1998 (1998) no. 5, pp. 225-252 | DOI | MR | Zbl
[17] Generalizing Tanisaki’s ideal via ideals of truncated symmetric functions, J. Algebraic Combin., Volume 37 (2013) no. 1, pp. 167-199 | DOI | MR | Zbl
[18] Affine pavings of Hessenberg varieties for semisimple groups, Selecta Math. (N.S.), Volume 19 (2013) no. 4, pp. 903-922 | DOI | MR | Zbl
[19] The connectedness of Hessenberg varieties, J. Algebra, Volume 437 (2015), pp. 34-43 | DOI | MR | Zbl
[20] The Betti numbers of regular Hessenberg varieties are palindromic, Transform. Groups, Volume 23 (2018) no. 2, pp. 491-499 | DOI | MR | Zbl
[21] Basic graph theory, Undergraduate Topics in Computer Science, Springer, Cham, 2017, x+169 pages | DOI | MR | Zbl
[23] Exponents for
[24] http://front.math.ucdavis.edu/math.SG/0211231, personal communication) | Numdam
([25] A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math., Volume 111 (1995) no. 1, pp. 166-194 | DOI | MR | Zbl
[26] Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR | Zbl
[27] On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A, Volume 62 (1993) no. 2, pp. 261-279 | DOI | MR | Zbl
[28] The Hessenberg representation, Ph. D. Thesis, The University of Iowa (USA) (2013) | MR
[29] Linear conditions imposed on flag varieties, Amer. J. Math., Volume 128 (2006) no. 6, pp. 1587-1604 | DOI | MR | Zbl
[30] Permutation actions on equivariant cohomology of flag varieties, Toric topology (Contemp. Math.), Volume 460, Amer. Math. Soc., Providence, RI, 2008, pp. 365-384 | DOI | MR | Zbl
Cité par Sources :