Logarithmic concavity of Schur and related polynomials

Huh, June; Matherne, Jacob P.; Mészáros, Karola; Dizier, Avery St.

doi:10.48550/arxiv.1906.09633

Cited by 3 publications

(12 citation statements)

References 16 publications

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“…As a final application we show how knowing that Schur classes of nef bundles lie in HR(X) gives another proof of a result of Huh-Matherne-Mészáros-Dizier [12] that the normalized Schur polynomials are Lorentzian.…”

mentioning

confidence: 85%

“…Lorentzian Property of Schur polynomials. We end with a discussion on how our results relate to those of Huh-Matherne-Mészáros-Dizier [12]. To do so we need some definitions that come from [3].…”

Section: Combinations Of Derived Schur Classesmentioning

confidence: 95%

“…Although there are many places in which log-convexity and Schur polynomials meet (e.g. [4,9,12,14,18,19]) we are not aware of any previous inequalities that cover precisely those studied here. 1.2.…”

Section: Comparison With Previous Workmentioning

confidence: 99%

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On Hodge-Riemann Cohomology Classes

Ross,

Toma

2021

Preprint

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We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian.

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confidence: 85%

Section: Combinations Of Derived Schur Classesmentioning

confidence: 95%

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On Hodge-Riemann Cohomology Classes

Ross,

Toma

2021

Preprint

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“…While there have been recent breakthroughs in the degree of Grothendieck polynomials [24,28,29], much less is known about the structure of the support. The support has previously been conjecturally connected to generalized permutahedra via flow polytopes [25,Conjecture 5.1], and via the Lorentzian property [15,Conjecture 22]. In this paper, we give a new poset theoretic perspective on the support of any Grothendieck polynomial.…”

Section: Introductionmentioning

confidence: 98%

“…The lowest degree component of G w is the Schubert polynomial S w . Schubert polynomials have many combinatorial constructions and are well-understood [1,2,6,7,9,11,12,15,18,20,21,26,32]. However there is not nearly as much known combinatorially or discrete-geometrically about Grothendieck polynomials.…”

Section: Introductionmentioning

confidence: 99%

On the support of Grothendieck polynomials

Mészáros¹,

Setiabrata²,

Dizier³

2022

Preprint

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Grothendieck polynomials Gw of permutations w ∈ Sn were introduced by Lascoux and Schützenberger in 1982 as a set of distinguished representatives for the K-theoretic classes of Schubert cycles in the K-theory of the flag variety of C n . We conjecture that the exponents of nonzero terms of the Grothendieck polynomial Gw form a poset under componentwise comparison that is isomorphic to an induced subposet of Z n . When w ∈ Sn avoids a certain set of patterns, we conjecturally connect the coefficients of Gw with the Möbius function values of the aforementioned poset with 0 appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations.

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Complements of Schubert polynomials

Fan¹,

Guo²,

Liu³

2020

Preprint

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Let S w (x) = S w (x 1 , . . . , x n ) be the Schubert polynomial for a permutation w of {1, 2, . . . , n}. For a composition µ = (µ 1 , . . . , µ n ) ∈ Z n ≥0 , writewith respect to µ. Huh, Matherne, Mészáros and St. Dizier proved that N() is a Lorentzian polynomial, where N is a linear operator sending a monomialThey further conjectured that N(S w (x)) is Lorentzian. Motivated by this conjecture, we investigate the problem when x µ S w (x −1 ) is still a Schubert polynomial. If x µ S w (x −1 ) is a Schubert polynomial, then N(S w (x)) will be Lorentzian. In this paper, we pay attention to the typical case that µ = δ n = (n − 1, . . . , 1, 0) is the staircase partition. Our result shows that x δn S w (x −1 ) is a Schubert polynomial if and only if w avoids the two patterns 132 and 312.

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Logarithmic concavity of Schur and related polynomials

Abstract: We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the special case of Kostka numbers.

Cited by 3 publications

References 16 publications

On Hodge-Riemann Cohomology Classes

On Hodge-Riemann Cohomology Classes

On the support of Grothendieck polynomials

Complements of Schubert polynomials

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