Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley

Hamaker, Zachary; Pechenik, Oliver; Speyer, David E; Weigandt, Anna

doi:10.5802/alco.93

Cited by 6 publications

(7 citation statements)

References 6 publications

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“…Remark Our convention for the indexing of Schubert polynomials differs from that often used in the literature by an inverse in the subscript. However, our convention agrees with that used in [3, 5, 6]. …”

Section: Background and Conventionsmentioning

confidence: 66%

“…Proposition 4.1 below indicates a connection between Schubert polynomials and the weak‐order weights appearing in Section 3. Let

\nabla = \sum_{i = 1}^{n} \partial / \partial x_{i}

; the following proposition was the key observation of [6]: Proposition Let

σ \in S_{n}

, then

\nabla {frakturS}_{σ} = \sum_{σ s_{i} ⋖ σ} i {frakturS}_{σ s_{i}} .

…”

Section: Strong‐order Macdonald Identities For Schubert Polynomialsmentioning

confidence: 99%

“…Hamaker, Pechenik, Speyer, and Weigandt [6] used this result to give a simple new proof of Macdonald's celebrated reduced word formula for the principal specialization of Schubert polynomials; we adapt this proof technique in the proof of Theorem 1.5 in Section 4.2. Theorem Let

σ \in S_{n}

.…”

Section: Strong‐order Macdonald Identities For Schubert Polynomialsmentioning

confidence: 99%

“…Björner [1] asked whether the weak order on the whole symmetric group

S_{n} = S_{n}^{\emptyset}

had the (strong) Sperner property; it was conjectured by Stanley [16] that it does, and this was proven in [4] by establishing a certain action of the Lie algebra

{sl}_{2}

which respects the weak order. This technique was developed further in [3, 5, 6] in order to establish new formulas for principal evaluations of Schubert polynomials. In this paper, we generalize this

{sl}_{2}

action (Theorem 1.2) in order to establish the strong Sperner property (Corollary 1.3) for all 132‐avoiding intervals (that is, intervals in the weak order below 132‐avoiding permutations), and in particular on all parabolic quotients

S_{n}^{J}

of the symmetric group.…”

Section: Introductionmentioning

confidence: 99%

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The Sperner property for 132‐avoiding intervals in the weak order

Gaetz

Tung

2020

Bull. London Math. Soc.

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A well-known result of Stanley from 1980 implies that the weak order on a maximal parabolic quotient of the symmetric group Sn has the Sperner property; this same property was recently established for the weak order on all of Sn by Gaetz and Gao, resolving a long-open problem. In this paper, we interpolate between these results by showing that the weak order on any parabolic quotient of Sn (and more generally on any 132-avoiding interval) has the Sperner property.This result is proven by exhibiting an action of sl2 respecting the weak order on these intervals. As a corollary we obtain a new formula for principal specializations of Schubert polynomials. Our formula can be seen as a strong Bruhat order analogue of Macdonald's reduced word formula. This proof technique and formula generalize work of Hamaker-Pechenik-Speyer-Weigandt and Gaetz-Gao.

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Section: Background and Conventionsmentioning

confidence: 66%

“…Proposition 4.1 below indicates a connection between Schubert polynomials and the weak‐order weights appearing in Section 3. Let

\nabla = \sum_{i = 1}^{n} \partial / \partial x_{i}

; the following proposition was the key observation of [6]: Proposition Let

σ \in S_{n}

, then

\nabla {frakturS}_{σ} = \sum_{σ s_{i} ⋖ σ} i {frakturS}_{σ s_{i}} .

…”

Section: Strong‐order Macdonald Identities For Schubert Polynomialsmentioning

confidence: 99%

σ \in S_{n}

.…”

Section: Strong‐order Macdonald Identities For Schubert Polynomialsmentioning

confidence: 99%

“…Björner [1] asked whether the weak order on the whole symmetric group

S_{n} = S_{n}^{\emptyset}

had the (strong) Sperner property; it was conjectured by Stanley [16] that it does, and this was proven in [4] by establishing a certain action of the Lie algebra

{sl}_{2}

{sl}_{2}

S_{n}^{J}

of the symmetric group.…”

Section: Introductionmentioning

confidence: 99%

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The Sperner property for 132‐avoiding intervals in the weak order

Gaetz

Tung

2020

Bull. London Math. Soc.

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“…where S w denotes the Schubert polynomial indexed by w. These polynomials are certain distinguished representatives of Schubert classes and were introduced by Lascoux and Schützenberger. Note that the left-hand side counts the number of reduced pipe dreams for w. We note here that Macdonald's reduced word identity has already been given several proofs [6,17,20], each shedding new light. More importantly for us, a q-analogue (conjectured by Macdonald) was first proved by Fomin and Stanley [15] by working in the NilCoxeter algebra; see also [31].…”

Section: Introductionmentioning

confidence: 95%

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

Nadeau,

Tewari

2021

Preprint

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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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