Schubert polynomials as integer point transforms of generalized permutahedra

Fink, Alex; Mészáros, Karola; Dizier, Avery St.

doi:10.1016/j.aim.2018.05.028

Cited by 41 publications

(56 citation statements)

References 16 publications

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“…Our main theorem is the following. [6]. The following conjecture, discovered jointly with Alex Fink, is a strengthening of [14, Conjecture 5.5] based on the results of this paper.…”

Section: Newton Polytopes Of Schubert and Grothendieck Polynomialssupporting

confidence: 61%

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From generalized permutahedra to Grothendieck polynomials via flow polytopes

Mészáros¹,

Dizier²

2020

Algebraic Combinatorics

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We study a family of dissections of flow polytopes arising from the subdivision algebra. To each dissection of a flow polytope, we associate a polynomial, called the left-degree polynomial, which we show is invariant of the dissection considered (proven independently by Grinberg). We prove that left-degree polynomials encode integer points of generalized permutahedra. Using that certain left-degree polynomials are related to Grothendieck polynomials, we resolve special cases of conjectures by Monical, Tokcan, and Yong regarding the saturated Newton polytope property of Grothendieck polynomials.

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“…Our main theorem is the following. [6]. The following conjecture, discovered jointly with Alex Fink, is a strengthening of [14, Conjecture 5.5] based on the results of this paper.…”

Section: Newton Polytopes Of Schubert and Grothendieck Polynomialssupporting

confidence: 61%

“…Newton (L G,F (t)) . (6) We first study the polytope Newton(L G (t)), and then the component pieces Newton (L G,F (t)). To start, we define a new constraint array.…”

Section: Newton Polytopes Of Left-degree Polynomialsmentioning

confidence: 99%

From generalized permutahedra to Grothendieck polynomials via flow polytopes

Mészáros¹,

Dizier²

2020

Algebraic Combinatorics

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“…In [45,Conjecture 5.5] it was conjectured that the Schubert polynomials have SNP property and they even conjectured a set of defining inequalities for the Newton polytope in [45,Conjecture 5.13]. A. Fink, K. Mézáros, and A. St. Dizier confirmed the full conjecture in [15]. As noted by the authors of [26] the combination of Proposition 5.1 (they use the equivalent [3,Corollary 10.2]) and [43,Theorem 15.40] (which is also included in [26,Theorem 6]) is enough to give an alternative proof to [45,Conjecture 5.5].…”

Section: Applicationsmentioning

confidence: 93%

When are multidegrees positive?

Castillo

Cid-Ruiz

et al. 2020

Advances in Mathematics

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be a multiprojective space over k, and X ⊆ P be a closed subscheme of P. We provide necessary and sufficient conditions for the positivity of the multidegrees of X. As a consequence of our methods, we show that when X is irreducible, the support of multidegrees forms a discrete algebraic polymatroid. In algebraic terms, we characterize the positivity of the mixed multiplicities of a standard multigraded algebra over an Artinian local ring, and we apply this to the positivity of mixed multiplicities of ideals. Furthermore, we use our results to recover several results in the literature in the context of combinatorial algebraic geometry.

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“…We now work towards proving Theorem 5.8. We start by reviewing some material from [4] for the reader's convenience. We then derive several lemmas that simplify the proof of Theorem 5.8.…”

Section: A Coefficient-wise Inequality For Dual Characters Of Flaggedmentioning

confidence: 99%

“…Knutson and Miller also showed them to be multidegrees of matrix Schubert varieties [7]. There are a number of combinatorial formulas for the Schubert polynomials [1,2,5,6,9,12,14,17], yet only recently has the structure of their supports been investigated: the support of a Schubert polynomial S w is the set of all integer points of a certain generalized permutahedron P(w) [4,15]. The question motivating this paper is to characterize when S w equals the integer point transform of P(w), in other words, when all the coefficients of S w are equal to 0 or 1.…”

Section: Introductionmentioning

confidence: 99%

Zero-one Schubert polynomials

2020

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We prove that if $$\sigma \in S_m$$ σ ∈ S m is a pattern of $$w \in S_n$$ w ∈ S n , then we can express the Schubert polynomial $$\mathfrak {S}_w$$ S w as a monomial times $$\mathfrak {S}_\sigma $$ S σ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar’s orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on $$\mathfrak {S}_w$$ S w being zero-one. In this case, the Schubert polynomial $$\mathfrak {S}_w$$ S w is equal to the integer point transform of a generalized permutahedron.

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Schubert polynomials as integer point transforms of generalized permutahedra

Cited by 41 publications

References 16 publications

From generalized permutahedra to Grothendieck polynomials via flow polytopes

From generalized permutahedra to Grothendieck polynomials via flow polytopes

When are multidegrees positive?

Zero-one Schubert polynomials

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