Abstract
We prove that if sigma in S_m is a pattern of w in S_n, then we can express the Schubert polynomial mathfrak {S}_w as a monomial times mathfrak {S}_sigma (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 being zero-one. In this case, the Schubert polynomial mathfrak {S}_w is equal to the integer point transform of a generalized permutahedron.
Highlights
Schubert polynomials, introduced by Lascoux and Schützenberger in [10], represent cohomology classes of Schubert cycles in the flag variety
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]
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
Summary
Schubert polynomials, introduced by Lascoux and Schützenberger in [10], represent cohomology classes of Schubert cycles in the flag variety. 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). 4 that multiplicity-freeness is a necessary condition for Sw to be zero-one In the latter proof we assume Theorem 1.2, whose generalization (Theorem 5.8) and proof is the subject of Sect. In the latter proof we assume Theorem 1.2, whose generalization (Theorem 5.8) and proof is the subject of Sect. 5
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