The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A

Abstract

Abstract. Let n be a fixed positive integer and h : {1, 2, . . . , n} → {1, 2, . . . , n} a Hessenberg function. The main results of this paper are twofold. First, we give a systematic method, depending in a simple manner on the Hessenberg function h, for producing an explicit presentation by generators and relations of the cohomology ring H * (Hess(N, h)) with Q coefficients of the corresponding regular nilpotent Hessenberg variety Hess(N, h). Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring H * (Hess(N, h)) of the regular nilpotent Hessenberg variety and the Sn-invariant subring H * (Hess(S, h)) Sn of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the Sn-action on H * (Hess(S, h)) defined by Tymoczko). Our second main result implies that dim Q H k (Hess(N, h)) = dim Q H k (Hess(S, h)) Sn for all k and hence partially proves the Shareshian-Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley-Stembridge conjecture. A proof of the full Shareshian-Wachs conjecture was recently given by Brosnan and Chow, but in our special case, our methods yield a stronger result (i.e. an isomorphism of rings) by more elementary considerations. This paper provides detailed proofs of results we recorded previously in a research announcement.

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

“…The Shareshian-Wachs conjecture was proved in 2015 by Brosnan and Chow [3] (also independently by Guay-Paquet [10]) by showing a remarkable relationship between the Betti numbers of different Hessenberg varieties. (Direct computations of cohomology rings of certain Hessenberg varieties also yield partial proofs of the Shareshian-Wachs conjecture; see [1,2].) It then follows that, in order to prove the graded Stanley-Stembridge conjecture, it suffices to prove that the cohomology H 2i (Hess(S, h)) for each i is a non-negative combination of the tabloid representations M λ [6, Part II, Section 7.2] of S n for λ a partition of n. In other words, given the decomposition (1.1) H 2i (Hess(S, h)) = λ⊢n c λ,i M λ in the representation ring Rep(S n ) of S n , the coefficients c λ,i are non-negative.…”

Hessenberg varieties, Slodowy slices, and integrable systems

“…Note also that T * X carries a canonical symplectic form, with respect to which the cotangent lift action is Hamiltonian. One can define a moment map µ : T * X → g * by the property (5) (µ(x, γ))(z) = γ(z(x)), x ∈ X, γ ∈ T * x X, z ∈ g. Our discussion now turns to Poisson-geometric considerations, for which we let X be a smooth algebraic variety with structure sheaf O X . One calls X a Poisson variety if O X has been enriched to a sheaf of Poisson algebras, (O X , {·, ·}).…”

“…It follows that the cotangent lifts of (32a) and (32b) are commuting Hamiltonian actions of G on T * G. We shall let µ L : T * G → g and µ R : T * G → g denote the moment maps for these respective cotangent lifts (see (5)).…”

Geometry of Regular Hessenberg Varieties

“…A regular Hessenberg variety is the one defined by a regular element x in g, namely an element x ∈ g whose Lie algebra centralizer Z g (x) has the minimum possible dimension. There are two cases of regular Hessenberg varieties that are well-studied: (1) if s ∈ g is a regular semisimple element, then X(s, H) is called a regular semisimple Hessenberg variety; (2) if N 0 ∈ g is a regular nilpotent element, then X(N 0 , H) is called a regular nilpotent Hessenberg variety. For example, one can choose a Hessenberg space H so that X(s, H) is a toric variety and X(N 0 , H) is the Peterson variety (see Section 2.1 for details).…”