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Register a new userA filtration on the cohomology rings of regular nilpotent Hessenberg varieties
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Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincare polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of Hessenberg Schubert polynomials in the context of regular nilpotent Hessenberg varieties, and outline several open questions pertaining to them.
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Let $n$ be a fixed positive integer and $h: {1,2,ldots,n} rightarrow {1,2,ldots,n}$ 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 $H^ast(Hess(mathsf{N},h))$ with $mathbb{Q}$ coefficients of the corresponding regular nilpotent Hessenberg variety $Hess(mathsf{N},h)$. 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 $H^*(Hess(mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $S_n$-invariant subring $H^*(Hess(mathsf{S},h))^{S_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $S_n$-action on $H^*(Hess(mathsf{S},h))$ defined by Tymoczko). Our second main result implies that $mathrm{dim}_{mathbb{Q}} H^k(Hess(mathsf{N},h)) = mathrm{dim}_{mathbb{Q}} H^k(Hess(mathsf{S},h))^{S_n}$ 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.
In this paper we construct an additive basis for the cohomology ring of a regular nilpotent Hessenberg variety which is obtained by extending all Poincare duals of smaller regular nilpotent Hessenberg varieties. In particular, all of the Poincare duals of smaller regular nilpotent Hessenberg varieties in the given regular nilpotent Hessenberg variety are linearly independent.
We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form $h=(h(1),ndots,n)$ in Lie type $A_{n-1}$. The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borels presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group $mathfrak{S}_n$ on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the $mathfrak{S}_n$-invariant subring of the cohomology ring.
Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet connects the well-known Stanley-Stembridge conjecture in combinatorics to the dot action of the symmetric group $S_n$ on the cohomology rings $H^*(Hess(S,h))$ of regular semisimple Hessenberg varieties. In particular, in order to prove the Stanley-Stembridge conjecture, it suffices to construct (for any Hessenberg function $h$) a permutation basis of $H^*(Hess(S,h))$ whose elements have stabilizers isomorphic to Young subgroups. In this manuscript we give several results which contribute toward this goal. Specifically, in some special cases, we give a new, purely combinatorial construction of classes in the $T$-equivariant cohomology ring $H^*_T(Hess(S,h))$ which form permutation bases for subrepresentations in $H^*_T(Hess(S,h))$. Moreover, from the definition of our classes it follows that the stabilizers are isomorphic to Young subgroups. Our constructions use a presentation of the $T$-equivariant cohomology rings $H^*_T(Hess(S,h))$ due to Goresky, Kottwitz, and MacPherson. The constructions presented in this manuscript generalize past work of Abe-Horiguchi-Masuda, Chow, and Cho-Hong-Lee.
Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $mathcal{A}_I$, the regular nilpotent Hessenberg variety $mbox{Hess}(N,I)$, and the regular semisimple Hessenberg variety $mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $mathcal{A}_I$ is isomorphic to $H^*(mbox{Hess}(N,I))$ and $H^*(mbox{Hess}(S,I))^W$, the invariants in $H^*(mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, and generalizes Borels celebrated theorem showing that the coinvariant algebra of $W$ is isomorphic to the cohomology ring of the flag variety $G/B$. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map $H^*(G/B)to H^*(mbox{Hess}(N,I))$ announced by Dale Peterson and an affirmative answer to a conjecture of Sommers-Tymoczko are immediate consequences. We also give an explicit ring presentation of $H^*(mbox{Hess}(N,I))$ in types $B$, $C$, and $G$. Such a presentation was already known in type $A$ or when $mbox{Hess}(N,I)$ is the Peterson variety. Moreover, we find the volume polynomial of $mbox{Hess}(N,I)$ and see that the hard Lefschetz property and the Hodge-Riemann relations hold for $mbox{Hess}(N,I)$, despite the fact that it is a singular variety in general.
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