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Zero-one Schubert polynomials. (English) Zbl 1464.14057
Schubert polynomials \(\mathfrak{S}_w\) represent cohomology classes of Schubert cycles in the full flag variety. Their coefficients are nonnegative integers. There exists a number of combinatorial formulas for computing these coefficients. This paper is devoted to the following question: when are all the coefficients of a Schubert polynomial equal to \(0\) or \(1\)? Such polynomials are called zero-one Schubert polynomials.
To answer this question, the authors first make the following observation: if a permutation \(\sigma\in S_m\) is a pattern of \(w\in S_n\), then the Schubert polynomial \(\mathfrak{S}_w\) equals a monomial times \(\mathfrak{S}_\sigma\) plus a polynomial with nonnegative coefficients. Hence the set of 0-1 Schubert polynomials is closed under pattern containment. Using Magyar’s orthodontia, an inductive algorithm for computing \(\mathfrak{S}_w\) in terms of the Rothe diagram of \(w\), they describe the set of twelve avoided patterns, and also formulate equivalent confitions for a Schubert polynomial to be 0-1 in terms of Rothe diagrams and orthodontic sequences of permutations.
According to the recent result of the same authors about the supports of Schubert polynomials (see [A. Fink et al., Adv. Math. 332, 465–475 (2018; Zbl 1443.05179)]), this implies that each 0-1 Schubert polynomial is equal to the integer transform of a generalized permutahedron.

14N15 Classical problems, Schubert calculus
14M15 Grassmannians, Schubert varieties, flag manifolds
05E14 Combinatorial aspects of algebraic geometry
05A05 Permutations, words, matrices
Zbl 1443.05179
Full Text: DOI
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