The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture

Harada, Megumi; Precup, Martha E.

doi:10.5802/alco.76

Harada, Megumi ¹; Precup, Martha E.²

¹ Department of Mathematics and Statistics McMaster University 1280 Main Street West Hamilton Ontario L8S4K1, Canada
² Department of Mathematics and Statistics Washington University in St. Louis One Brookings Drive St. Louis Missouri 63130, U.S.A.

Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1059-1108.

Abstract

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 $𝔖_{n}$ -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.

Received: 2018-07-29
Revised: 2019-03-01
Accepted: 2019-03-18
Published online: 2019-12-03

MR Zbl

DOI: 10.5802/alco.76

Classification: 14M17, 05E05
Keywords: Stanley–Stembridge conjecture, symmetric functions, e-positivity, Hessenberg varieties, abelian ideal

Author's affiliations:

Harada, Megumi ¹; Precup, Martha E. ²

¹ Department of Mathematics and Statistics McMaster University 1280 Main Street West Hamilton Ontario L8S4K1, Canada
² Department of Mathematics and Statistics Washington University in St. Louis One Brookings Drive St. Louis Missouri 63130, U.S.A.

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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. https://alco.centre-mersenne.org/articles/10.5802/alco.76/

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