[Paper Review] The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture

Megumi Harada, Martha Precup|arXiv (Cornell University)|Sep 20, 2017

Advanced Combinatorial Mathematics15 references21 citations

TL;DR

This paper establishes an inductive formula for the Sn-representation on the cohomology of abelian regular semisimple Hessenberg varieties via Tymoczko's dot action, proving that the graded Stanley-Stembridge conjecture holds in the abelian case. The key result is a recursive decomposition of cohomology classes in terms of smaller Hessenberg varieties, showing that all coefficients in the tabloid representation decomposition are non-negative integers.

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.

Motivation & Objective

• . The research aims to prove the graded Stanley-Stembridge conjecture for a specific class of Hessenberg varieties.
• The problem is to show that the cohomology of regular semisimple Hessenberg varieties decomposes into non-negative combinations of tabloid representations of the symmetric group Sn.
• The objective is to establish an inductive formula for the coefficients in the Sn-representation decomposition of the cohomology ring.
• The study focuses on abelian Hessenberg varieties, defined via abelian ideals in gl(n,C), to achieve this decomposition.
• The goal is to generalize prior results by Shareshian-Wachs and Teff using geometric and combinatorial techniques on acyclic orientations and sink sets.

Proposed method

• . The authors define abelian Hessenberg varieties using Hessenberg functions h such that the associated ideal Ih is abelian in the Lie algebra gl(n,C).
• They use Tymoczko's dot action to define an Sn-action on the cohomology of regular semisimple Hessenberg varieties.
• The method relies on decomposing acyclic orientations of the incomparability graph Γh by their sink sets, particularly maximal ones.
• An inductive formula is constructed by relating orientations on Γh to those on smaller graphs ΓhT via subsets T ∈ SK2(Γh), the set of maximal 2-sink sets.
• The Poincaré polynomials of regular abelian Hessenberg varieties are computed inductively using degree shifts deg(T) associated with each T.
• The proof proceeds by induction on n, using the fact that if h is abelian, then the associated hT for each T ∈ SK2(Γh) is also abelian, preserving the inductive hypothesis.

Experimental results

Research questions

• RQ1 . Does the cohomology of abelian regular semisimple Hessenberg varieties decompose into non-negative combinations of tabloid representations Mλ of Sn?
• RQ2 . Can an inductive formula be constructed for the coefficients cλ,i in the decomposition H2i(Hess(S,h)) = ∑λ⊢n cλ,i Mλ?
• RQ3 . How do maximal sink sets in the incomparability graph Γh relate to the structure of the cohomology representation?
• RQ4 . Is there a generalization of the inductive formula beyond the abelian case, and what are the limitations of such a generalization?
• RQ5 . What is the role of the height of an ideal and the structure of acyclic orientations in determining the cohomology representation?

Key findings

• . The paper proves that for abelian Hessenberg varieties, the cohomology H2i(Hess(S,h)) is a non-negative linear combination of tabloid representations Mλ of Sn.
• . An inductive formula is established: H2i(Hess(S,h)) = c(n),iM(n) + ∑T∈SK2(Γh) (∑μ⊢(n−2) cTμ,i−deg(T) M(μ1+1,μ2+1)), where coefficients are inherited from smaller Hessenberg varieties.
• . The coefficients cλ,i are non-negative integers, confirming the graded Stanley-Stembridge conjecture in the abelian case.
• . The inductive structure is preserved: if h is abelian, then hT is also abelian for each T ∈ SK2(Γh), enabling the inductive proof.
• . The Poincaré polynomials of regular abelian Hessenberg varieties are computed inductively using degree shifts deg(T) associated with maximal 2-sink sets.
• . A conjecture is proposed (Conjecture 8.1) extending the inductive formula to general Hessenberg varieties, but it only determines coefficients for partitions λ with m(Γh) parts, not the full representation.