The Betti numbers of regular Hessenberg varieties are palindromic

Precup, Martha

doi:10.48550/arxiv.1603.07662

Cited by 3 publications

(7 citation statements)

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“…On the other hand, using her generalization [37] of Tymoczko's computation of the Betti numbers of Hessenberg varieties in type A, Precup generalized our palindromicity results (Corollaries 36 and 37) to regular Hessenberg varieties for arbitrary complex semi-simple Lie algebras [36]. As it happens, our proof of palindromicity is rather indirect, relying in an essential way on the chromatic quasisymmetric function and a palindromicity theorem for that function proved by Shareshian and Wachs ([44,Corollary 4.6]).…”

Section: Later Workmentioning

confidence: 91%

Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties

Brosnan

Chow

2018

Advances in Mathematics

152

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Motivated by a 1993 conjecture of Stanley and Stembridge, Shareshian and Wachs conjectured that the characteristic map takes the character of the dot action of the symmetric group on the cohomology of a regular semisimple Hessenberg variety to ωXG(t), where XG(t) is the chromatic quasisymmetric function of the incomparability graph G of the corresponding natural unit interval order, and ω is the usual involution on symmetric functions. We prove the Shareshian-Wachs conjecture.Our proof uses the local invariant cycle theorem of Beilinson-Bernstein-Deligne to obtain a surjection, which we call the local invariant cycle map, from the cohomology of a regular Hessenberg variety of Jordan type λ to a space of local invariant cycles. As λ ranges over all partitions, the local invariant cycles collectively contain all the information about the dot action on a regular semisimple Hessenberg variety. We then prove a result showing that, under suitable hypotheses, the local invariant cycle map is an isomorphism if and only if the special fiber has palindromic cohomology. (This is a general theorem, which independent of the Hessenberg variety context.) Applying this result to the universal family of Hessenberg varieties, we show that, in our case, the surjections are actually isomorphisms, thus reducing the Shareshian-Wachs conjecture to computing the cohomology of a regular Hessenberg variety. But this cohomology has already been described combinatorially by Tymoczko, and, using a new reciprocity theorem for certain quasisymmetric functions, we show that Tymoczko's description coincides with the combinatorics of the chromatic quasisymmetric function.

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Section: Later Workmentioning

confidence: 91%

Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties

Brosnan

Chow

2018

Advances in Mathematics

152

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“…One can see dim C Hess(N, I) = |I| from the above formula but it is also a special case of [32,Corollary 2.7]. The palindromicity of the Poincaré polynomial of Hess(N,…”

Section: It Follows Thatmentioning

confidence: 92%

“…Proof. We shall briefly explain how the theorem follows from results in [31] and [32]. Let X w = BwB/B be the Schubert cell of the flag variety G/B associated to w ∈ W .…”

Section: Settingmentioning

confidence: 99%

“…The following is a summary of some results from [31,32] about Hess(N, I) ([45] in the classical types). Theorem 3.3 ([31,32]).…”

Section: Settingmentioning

confidence: 99%

“…The following is a summary of some results from [31,32] about Hess(N, I) ([45] in the classical types). Theorem 3.3 ([31,32]). The regular nilpotent Hessenberg variety Hess(N, I) has no odd degree cohomology and is of dimension equal to |I|.…”

Section: Settingmentioning

confidence: 99%

See 2 more Smart Citations

Hessenberg varieties and hyperplane arrangements

Abe

Horiguchi

Masuda

et al. 2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

104

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AbstractGiven 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 {\operatorname{Hess}(N,I)}, and the regular semisimple Hessenberg variety {\operatorname{Hess}(S,I)}. We show that a certain graded ring derived from the logarithmic derivation module of {\mathcal{A}_{I}} is isomorphic to {H^{*}(\operatorname{Hess}(N,I))} and {H^{*}(\operatorname{Hess}(S,I))^{W}}, the invariants in {H^{*}(\operatorname{Hess}(S,I))} under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel’s 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^{*}(\operatorname{Hess}(N,I))} announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of {H^{*}(\operatorname{Hess}(N,I))} in types B, C, and G. Such a presentation was already known in type A and when {\operatorname{Hess}(N,I)} is the Peterson variety. Moreover, we find the volume polynomial of {\operatorname{Hess}(N,I)} and see that the hard Lefschetz property and the Hodge–Riemann relations hold for {\operatorname{Hess}(N,I)}, despite the fact that it is a singular variety in general.

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Geometry of Regular Hessenberg Varieties

Abe

Fujita

2020

Transformation Groups

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Let g be a complex semisimple Lie algebra. For a regular element x in g and a Hessenberg space H ⊆ g, we consider a regular Hessenberg variety X(x, H) in the flag variety associated with g. We take a Hessenberg space so that X(x, H) is irreducible, and show that the higher cohomology groups of the structure sheaf of X(x, H) vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.

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The Betti numbers of regular Hessenberg varieties are palindromic

Cited by 3 publications

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Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties

Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties

Hessenberg varieties and hyperplane arrangements

Geometry of Regular Hessenberg Varieties

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