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Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams. (English) Zbl 1356.14039
The 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.
Reviewer: Cenap Özel (Bolu)

14M15 Grassmannians, Schubert varieties, flag manifolds
14N15 Classical problems, Schubert calculus
Full Text: DOI arXiv
[1] Bergeron, N.; Billey, S., RC-graphs and Schubert polynomials, Exp. Math., 2, 4, 257-269, (1993), MR 1281474 (95g:05107) · Zbl 0803.05054
[2] Billey, S. C.; Jockusch, W.; Stanley, R. P., Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., 2, 4, 345-374, (1993), MR 1241505 (94m:05197) · Zbl 0790.05093
[3] Demazure, M., Une nouvelle formule des caractères, Bull. Sci. Math. (2), 98, 3, 163-172, (1974), MR 0430001 (55 #3009) · Zbl 0365.17005
[4] Edelman, P.; Greene, C., Balanced tableaux, Adv. Math., 63, 1, 42-99, (1987), MR 871081 (88b:05012) · Zbl 0616.05005
[5] Eilenberg, S.; Mac Lane, S., On the groups of \(H(\operatorname{\Pi}, n)\). I, Ann. of Math. (2), 58, 55-106, (1953), MR 0056295 (15,54b)
[6] Fomin, S.; Stanley, R. P., Schubert polynomials and the nil-Coxeter algebra, Adv. Math., 103, 2, 196-207, (1994), MR 1265793 (95f:05115) · Zbl 0809.05091
[7] Gessel, I. M., Multipartite P-partitions and inner products of skew Schur functions, (Combinatorics and Algebra, Boulder, Colo., 1983, Contemp. Math., vol. 34, (1984), Amer. Math. Soc. Providence, RI), 289-317
[8] Hoffman, M. E., Quasi-shuffle products, J. Algebraic Combin., 11, 1, 49-68, (2000), MR 1747062 (2001f:05157) · Zbl 0959.16021
[9] Li, N., A canonical expansion of the product of two Stanley symmetric functions, J. Algebraic Combin., 39, 4, 833-851, (2014), MR 3199028 · Zbl 1292.05259
[10] Lascoux, A.; Schützenberger, M.-P., Polynômes de Schubert, C. R. Acad. Sci., Sér. 1 Math., 294, 13, 447-450, (1982), MR 660739 (83e:14039) · Zbl 0495.14031
[11] Lascoux, A.; Schützenberger, M.-P., Keys & standard bases, (Invariant Theory and Tableaux, Minneapolis, MN, 1988, IMA Vol. Math. Appl., vol. 19, (1990), Springer New York), 125-144, MR 1035493 (91c:05198) · Zbl 0815.20013
[12] Macdonald, I. G., Notes on Schubert polynomials, (1991), LACIM, Univ. Quebec a Montreal Montreal, PQ · Zbl 0784.05061
[13] Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, (1995), The Clarendon Press Oxford University Press New York, With contributions by A. Zelevinsky, Oxford Science Publications · Zbl 0487.20007
[14] Stanley, R. P., On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., 5, 4, 359-372, (1984), MR 782057 (86i:05011) · Zbl 0587.20002
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