Hessenberg varieties and hyperplane arrangements

Abe, Takuro; Horiguchi, Tatsuya; Masuda, Mikiya; Murai, Shinji; Sagawa, Takashi

doi:10.1515/crelle-2018-0039

Cited by 37 publications

(108 citation statements)

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“…We first introduce the volume polynomial of a subvariety of Flag(C n ) and then consider the special case when the subvariety is the regular semisimple Hessenberg variety. In this case, some of our previous results [2,1] links the volume polynomial of Hess(S, h) to the volume polynomial of the Gelfand-Zetlin polytope GZ(λ). Some explicit computations in small-n cases led us to believe that, firstly, the volume polynomial for Hess(S, h) should be an appropriate linear combination of the volumes of faces of GZ(λ).…”

Section: Volume Polynomials Of Regular Semisimple Hessenberg Varietiesmentioning

confidence: 98%

“…Remark 4.2. The discussion in [2] is given in the language of Poincaré duality algebras, and [2, Theorem 11.3] is not stated in exactly the form as given above. The volume polynomial as discussed in [2,Section 11] is the polynomial whose annihilator is equal to the ideal defining the cohomology ring H * (Hess(S, h)) of the regular semisimple Hessenberg variety; it is only determined up to a scalar multiple.…”

Section: Volume Polynomials Of Regular Semisimple Hessenberg Varietiesmentioning

confidence: 99%

“…[21,6,9,10]). Most recently, volume polynomials also appeared in the study of Hessenberg varieties through the work of the second and third authors together with Abe, Murai, and Sato [2], in which they use the theory of hyperplane arrangements to derive explicit generators and relations for the cohomology rings of regular nilpotent Hessenberg varieties in certain Lie types. In related work, the first author together with Abe, DeDieu, and Galetto studied the volume polynomials of both regular semisimple and regular nilpotent Hessenberg varieties in relation to the study of Newton-Okounkov bodies [1].…”

Section: Introductionmentioning

confidence: 99%

“…In related work, the first author together with Abe, DeDieu, and Galetto studied the volume polynomials of both regular semisimple and regular nilpotent Hessenberg varieties in relation to the study of Newton-Okounkov bodies [1]. It can be seen from the explicit formula for the volume polynomials of regular nilpotent Hessenberg varieties given in [2] that it is intimately related to the volume of Gelfand-Zetlin polytopes. Therefore, the time seemed ripe for a careful study of the volume polynomials of Hessenberg and related subvarieties.…”

Section: Introductionmentioning

confidence: 99%

“…, λ n . It turns out that the geometric volume polynomial Vol λ (Hess(S, h)) of a regular semisimple Hessenberg variety in Flag(C n ) can be obtained by taking certain derivatives, with respect to the λ i 's, of the combinatorial volume polynomial Vol(GZ(λ)) of GZ(λ) [1,2]. Straightforward computations in small-n cases lead us to believe that this geometric volume Vol λ (Hess(S, h)) can alternately be expressed as a non-negative linear combination of the combinatorial volumes of certain faces of GZ(λ).…”

Section: Introductionmentioning

confidence: 99%

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The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope

Harada

Horiguchi

Masuda

et al. 2019

Proc. Steklov Inst. Math.

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Regular semisimple Hessenberg varieties are subvarieties of the flag variety Flag(C n ) arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of Abe-Horiguchi-Masuda-Murai-Sato and Abe-DeDieu-Galetto-Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand-Zetlin polytope GZ(λ) for λ = (λ 1 , λ 2 , . . . , λn). The main results of this manuscript use and generalize tools developed by Anderson-Tymoczko, Kiritchenko-Smirnov-Timorin, and Postnikov, in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand-Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the α i := λ i − λ i+1 . In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial (n − 1)-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in Flag(C n ) as a sum of the cohomology classes of a certain set of Richardson varieties.

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Section: Volume Polynomials Of Regular Semisimple Hessenberg Varietiesmentioning

confidence: 98%

Section: Volume Polynomials Of Regular Semisimple Hessenberg Varietiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope

Harada

Horiguchi

Masuda

et al. 2019

Proc. Steklov Inst. Math.

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A Survey of Recent Developments on Hessenberg Varieties

Abe

Horiguchi

2020

Springer Proceedings in Mathematics &Amp; Statistics

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Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

Abe

DeDieu

Galetto

et al. 2018

Sel. Math. New Ser.

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In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton-Okounkov bodies of the two-dimensional Peterson variety.

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Hessenberg varieties and hyperplane arrangements

Abstract: AbstractGiven a semisimple complex linear algebraic group {{G}} and a lower ideal I in positive roots of G, three objects arise: t… Show more

Cited by 37 publications

References 50 publications

The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope

The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope

A Survey of Recent Developments on Hessenberg Varieties

Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

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