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 ${\U0001d516}_{n}$-representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko. Our result implies that a graded version of the Stanley–Stembridge conjecture holds in the abelian case, and generalizes results obtained by Shareshian–Wachs and Teff. Our proof uses previous work of Stanley, Gasharov, Shareshian–Wachs, and Brosnan–Chow, as well as results of the second author on the geometry and combinatorics of Hessenberg varieties. As part of our arguments, we obtain inductive formulas for the Poincaré polynomials of regular abelian Hessenberg varieties.

Revised:

Accepted:

Published online:

DOI: 10.5802/alco.76

Keywords: Stanley–Stembridge conjecture, symmetric functions, e-positivity, Hessenberg varieties, abelian ideal

^{1}; Precup, Martha E.

^{2}

@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, Volume 2 (2019) no. 6, pp. 1059-1108. doi : 10.5802/alco.76. http://www.numdam.org/articles/10.5802/alco.76/

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