## 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.

### MSC:

 14N15 Classical problems, Schubert calculus 14M15 Grassmannians, Schubert varieties, flag manifolds 05E14 Combinatorial aspects of algebraic geometry 05A05 Permutations, words, matrices

### Keywords:

Schubert polynomial; pattern avoidance; Rothe diagram

Zbl 1443.05179
Full Text:

### References:

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