The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials

Nadeau, Philippe; Tewari, Vasu

doi:10.48550/arxiv.2005.12194

Cited by 6 publications

(24 citation statements)

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“…These are in general hard to compute, and finding a combinatorial rule to describe them is a wide open problem. We approach understanding a w (q) from our explicit formula above and uncover some curious properties of these polynomials, extending previous work of the authors [34] in the case q = 1. We briefly describe some developments next, referring the reader to § §7.2, § §7.3, and § §7.4…”

Section: Introductionmentioning

confidence: 54%

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A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula

Nadeau,

Tewari

2021

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There is a striking similarity between Macdonald's reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety Permn as computed by Klyachko. Toward understanding this better, we undertake an in-depth study of a q-deformation of the Sn-invariant part of the rational cohomology ring of Permn, which we call the q-Klyachko algebra. We uncover intimate links between expansions in the basis of squarefree monomials in this algebra and various notions in algebraic combinatorics, thereby connecting seemingly unrelated results by finding a common ground to study them. Our main results are as follows.• A q-analog of divided symmetrization (q-DS) using Yang-Baxter elements in the Hecke algebra.It is a linear form that picks up coefficients in the squarefree basis. • A relation between q-DS and the ideal of quasisymmetric polynomials involving work of Aval-Bergeron-Bergeron. • A family of polynomials in q with nonnegative integral coefficients that specialize to Postnikov's mixed Eulerian numbers when q = 1. We refer to these new polynomials as remixed Eulerian numbers. For q > 0, their normalized versions occur as probabilities in the internal diffusion limited aggregation (IDLA) stochastic process. • A lift of Macdonald's reduced word identity in the q-Klyachko algebra.• The Schubert expansion of the Chow class of the standard split Deligne-Lusztig variety in type A, when q is a prime power.

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Section: Introductionmentioning

confidence: 54%

“…It follows easily from [34,36] that A c (1) = A c where the latter are the mixed Eulerian numbers introduced by Postnikov [37]. Furthermore, for c ∈ W r r , the following properties hold:…”

Section: Introductionmentioning

confidence: 99%

A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula

Nadeau,

Tewari

2021

Preprint

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“…Let R(v I ) denote the set of reduced words for v I . Following [Kly85,Kly95] (see also [NT21,Thm 8.1]), we have…”

Section: Corollary 52 Consider the Inclusionmentioning

confidence: 99%

“…This allows us to relate computations in the ordinary cohomology of the Peterson variety to corresponding computations on the ordinary cohomology of the permutahedral variety. In [Kly85,Kly85] (see also [NT21]), Klyachko presented a Giambelli formula expressing the pullback class j * σ w of any Schubert class as a polynomial in the divisor classes j * σ sα ,…”

Section: Introductionmentioning

confidence: 99%

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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“…Furthermore, these numbers are related to the intersection number of the toric variety associated with a weight polytope and the Schubert variety. We refer to [10,14,26,28,30].…”

Section: Introductionmentioning

confidence: 99%

Mixed Eulerian numbers and Peterson Schubert calculus

Horiguchi¹

2021

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In this paper we derive a combinatorial formula for mixed Eulerian numbers in type A from Peterson Schubert calculus. We also provide a simple computation for mixed Φ-Eulerian numbers in arbitrary Lie types. Contents 1. Introduction 1 2. Permutohedron 3 3. Equivariant cohomology of a permutohedral variety 4 4. Mixed Eulerian numbers 7 5. Computation for mixed Eulerian numbers and left-right diagrams 8 6. Mixed Φ-Eulerian numbers 13 7. Computation for mixed Φ-Eulerian numbers and Peterson Schubert calculus 15 Appendix A. A computation for m J i,K 28 References 35

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The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials

Cited by 6 publications

References 50 publications

A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula

A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula

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

Mixed Eulerian numbers and Peterson Schubert calculus

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