Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams.

*(English)*Zbl 1356.14039The Schubert polynomials give explicit polynomial representatives for the Schubert classes in the cohomology ring of the complete flag variety, with the goal of facilitating computations of intersection numbers. A. Lascoux and M.-P. Schützenberger [C. R. Acad. Sci., Paris, Sér. I 294, 447–450 (1982; Zbl 0495.14031)] first defined Schubert polynomials indexed by permutations in terms of divided difference operators, and later S. C. Billey et al. [J. Algebr. Comb. 2, No. 4, 345–374 (1993; Zbl 0790.05093)] and S. Fomin and R. P. Stanley [Adv. Math. 103, No. 2, 196–207 (1994; Zbl 0809.05091)] gave direct monomial expansions. N. Bergeron and S. Billey [Exp. Math. 2, No. 4, 257–269 (1993; Zbl 0803.05054)] reformulated this again to give a beautiful combinatorial definition of Schubert polynomials as generating functions for \(RC\)-graphs, often called pipe dreams.

In this paper, the authors introduce a new tool to aid in the study of Schubert polynomials.They define two new families of polynomials they call the monomial slide polynomials and fundamental slide polynomials. Both monomial and fundamental slide polynomials are combinatorially indexed by weak compositions, and both families form a basis of the polynomial ring. Moreover, the Schubert polynomials expand positively into the fundamental slide basis, which in turn expands positively into the monomial slide basis. While there are other bases that refine Schubert polynomials, most notably key polynomials, it has two main properties that make it a compelling addition to the theory of Schubert calculus. First, these polynomials exhibit a similar stability to that of Schubert polynomials, and so they facilitate a deeper understanding of the stable limit of Schubert polynomials, which, as originally shown by Macdonald, are Stanley symmetric functions. Second, and in sharp contrast to key polynomials, their bases themselves have positive structure constants, and so their Littlewood-Richardson rule for the fundamental slide expansion of a product of Schubert polynomials takes one step closer to giving a combinatorial formula for Schubert structure constants.

In this paper, the authors introduce a new tool to aid in the study of Schubert polynomials.They define two new families of polynomials they call the monomial slide polynomials and fundamental slide polynomials. Both monomial and fundamental slide polynomials are combinatorially indexed by weak compositions, and both families form a basis of the polynomial ring. Moreover, the Schubert polynomials expand positively into the fundamental slide basis, which in turn expands positively into the monomial slide basis. While there are other bases that refine Schubert polynomials, most notably key polynomials, it has two main properties that make it a compelling addition to the theory of Schubert calculus. First, these polynomials exhibit a similar stability to that of Schubert polynomials, and so they facilitate a deeper understanding of the stable limit of Schubert polynomials, which, as originally shown by Macdonald, are Stanley symmetric functions. Second, and in sharp contrast to key polynomials, their bases themselves have positive structure constants, and so their Littlewood-Richardson rule for the fundamental slide expansion of a product of Schubert polynomials takes one step closer to giving a combinatorial formula for Schubert structure constants.

Reviewer: Cenap Özel (Bolu)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14N15 | Classical problems, Schubert calculus |

##### Keywords:

Schubert polynomials; Stanley symmetric functions; pipe dreams; reduced decompositions; quasisymmetric functions##### References:

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