Mixed Eulerian numbers and Peterson Schubert calculus

Horiguchi, Tatsuya

doi:10.48550/arxiv.2104.14083

Cited by 3 publications

(9 citation statements)

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“…In our final results, we develop a Chevalley formula (Theorem 6.5), and a dual Monk rule (Theorem 6.7). A Monk rule for the ordinary cohomology of P was recently obtained by Horiguchi [Hor21]. The equivalence of the Chevalley formula and the Monk rule is a consequence of Equation (3) and Theorem 1.3.…”

Section: Introductionmentioning

confidence: 91%

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Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety

Goldin¹,

Singh²

2021

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We present a formula for the Poincaré dual in the flag manifold of the equivariant fundamental class of any regular nilpotent or regular semisimple Hessenberg variety as a polynomial in terms of certain Chern classes. We then develop a type-independent proof of the Giambelli formula for the Peterson variety, and use this formula to compute the intersection multiplicity of a Peterson variety with an opposite Schubert variety corresponding to a Coxeter word. Finally, we develop an equivariant Chevalley formula for the cap product of a divisor class with a fundamental class, and a dual Monk rule, for the Peterson variety.

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Section: Introductionmentioning

confidence: 91%

“…Drellich [Dre15] found a Giambelli formula for certain Coxeter elements (see Theorem 1.2) using a type-by-type analysis. Horiguchi [Hor21] obtained a Monk rule for ordinary cohomology using similar methods. In type A, an equivariant Monk rule was developed by Harada and Tymoczko [HT11].…”

Section: Introductionmentioning

confidence: 99%

Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety

Goldin¹,

Singh²

2021

Preprint

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“…Recall that we computed A c (q) for c = (0, 3, 0, 0, 0, 1, 3) in Example 5.4, and it is seen to be psu(26), which is in accordance with Theorem 5.6. Indeed, 7 2 − H(c) = 21 + 5 = 26.…”

Section: Degree Symmetry Unimodalitymentioning

confidence: 99%

“…(5) follows from the sum rule [17,Proposition 5.4], as explained in Remark 2.1. Finally, (7) is part of [17,Theorem 4.8].…”

Section: Introductionmentioning

confidence: 99%

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Remixed Eulerian numbers

Nadeau¹,

Tewari²

2022

Preprint

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Remixed Eulerian numbers are a polynomial q-deformation of Postnikov's mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as q-binomial coefficients and Garsia-Remmel's q-hit numbers. We study their combinatorics in more depth. As polynomials in q, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At q = 1, the former recovers Postnikov's interpretation whereas the latter recovers Liu's interpretation, both of which were obtained via methods different from ours. P. N is partially supported by the project ANR19-CE48-011-01. V. T. acknowledges the support from Simons Collaboration Grant #855592.

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A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

et al. 2020

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This manuscript is a contributed chapter in the forthcoming CRC Press volume, titled the Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety. The book, as a whole, is aimed at a diverse audience of researchers and graduate students seeking an expository introduction to the area. In our chapter, we give an overview of some of the past research on the cohomology rings of regular nilpotent Hessenberg varieties, with no claim to being exhaustive. For the purposes of this manuscript, we focus mainly on the case of Lie type A, with some brief remarks on the general Lie types. We end the chapter with a selection of topics currently active in this area.

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Mixed Eulerian numbers and Peterson Schubert calculus

Cited by 3 publications

References 20 publications

Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety

Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety

Remixed Eulerian numbers

A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

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