Zero-one Schubert polynomials
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Mathematische Zeitschrift
Zero-one Schubert polynomials Alex Fink1 · Karola Mészáros2,3 · Avery St. Dizier2 Received: 2 September 2019 / Accepted: 5 April 2020 © The Author(s) 2020
Abstract We prove that if σ ∈ Sm is a pattern of w ∈ Sn , then we can express the Schubert polynomial Sw as a monomial times Sσ (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 Sw being zero-one. In this case, the Schubert polynomial Sw is equal to the integer point transform of a generalized permutahedron.
1 Introduction Schubert polynomials, introduced by Lascoux and Schützenberger in [10], represent cohomology classes of Schubert cycles in the flag variety. 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 Sw is the set of all integer points of a certain generalized permutahedron P(w) [4,15]. The question motivating
Fink was partially supported by an Engineering and Physical Sciences Research Council grant (EP/M01245X/1), and is now supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 792432. Mészáros is partially supported by a National Science Foundation Grant (DMS 1501059), as well as by a von Neumann Fellowship at the IAS funded by the Fund for Mathematics and Friends of the Institute for Advanced Study.
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Alex Fink [email protected] Karola Mészáros [email protected] Avery St. Dizier [email protected]
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Alex Fink, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK
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Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
3
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
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this paper is to characterize when Sw equals the integer point transform of P(w), in other words, when all the coefficients of Sw are equal to 0 or 1. One of our main results is a pattern-avoidance characterization of the permutations corresponding to these polynomials: Theorem 1.1 The Schubert polynomial Sw is zero-one if and only if w avoids the patterns 12543, 13254, 13524, 13542, 21543, 125364, 125634, 215364, 215634, 315264, 315624, and 315642. In Theorem 4.8 we provide further equivalent conditions on the Schubert polynomial Sw being zero-one. One implication of Theorem 1.1 follows from our other main result, which relates the Schubert polynomials Sσ and Sw when σ is a pattern of w: Theorem 1.2 Fix w ∈ Sn and let σ ∈ Sn−1 be the pattern with Rothe diagram D(σ ) obtained by removing row k and column wk from D(w). Then Sw (x1 , . . . , xn )
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