Slide polynomials and subword complexes

E. Yu. Smirnov   ; A. A. Tutubalina

Sbornik. Mathematics, Tome 212 (2021) no. 10, pp. 1471-1490

Résumé

Subword complexes were defined by Knutson and Miller in 2004 to describe Gröbner degenerations of matrix Schubert varieties. Subword complexes of a certain type are called pipe dream complexes. The facets of such a complex are indexed by pipe dreams, or, equivalently, by monomials in the corresponding Schubert polynomial. In 2017 Assaf and Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes. The slide complexes appearing in such a way are shown to be homeomorphic to balls or spheres. For pipe dream complexes, such strata correspond to slide polynomials. Bibliography: 14 titles.

Détail

Keywords: flag varieties, Schubert polynomials, Grothendieck polynomials, simplicial complexes.

Bibliographie

[1] S. Assaf, D. Searles, “Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams”, Adv. Math., 306 (2017), 89–122 | DOI | MR | Zbl

[2] N. Bergeron, S. Billey, “RC-graphs and Schubert polynomials”, Experiment. Math., 2:4 (1993), 257–269 | DOI | MR | Zbl

[3] I. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand, “Schubert cells and cohomology of the spaces $G/P$”, Russian Math. Surveys, 28:3 (1973), 1–26 | DOI | MR | Zbl

[4] L. J. Billera, J. Scott Provan, “A decomposition property for simplicial complexes and its relation to diameters and shellings”, Second international conference on combinatorial mathematics (New York, 1978), Ann. New York Acad. Sci., 319, New York Acad. Sci., New York, 1979, 82–85 | DOI | MR | Zbl

[5] L. Escobar, K. Mészáros, “Subword complexes via triangulations of root polytopes”, Algebr. Comb., 1:3 (2018), 395–414 | DOI | MR | Zbl

[6] S. Fomin, A. N. Kirillov, “Grothendieck polynomials and the Yang–Baxter equation”, Formal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, DIMACS, Piscataway, NJ, 1994, 183–190 | MR

[7] S. Fomin, A. N. Kirillov, “The Yang–Baxter equation, symmetric functions, and Schubert polynomials” (Florence, 1993), Discrete Math., 153:1-3, Proceedings of the 5th conference on formal power series and algebraic combinatorics (1996), 123–143 | DOI | MR | Zbl

[8] A. Knutson, E. Miller, “Subword complexes in Coxeter groups”, Adv. Math., 184:1 (2004), 161–176 | DOI | MR | Zbl

[9] A. Knutson, E. Miller, “Gröbner geometry of Schubert polynomials”, Ann. of Math. (2), 161:3 (2005), 1245–1318 | DOI | MR | Zbl

[10] A. Lascoux, “Anneau de Grothendieck de la variété de drapeaux”, The Grothendieck Festschrift, v. III, Mod. Birkhäuser Class., 88, Birkhäuser/Springer, Cham, 2007, 1–34 | DOI | MR | Zbl

[11] A. Lascoux, M.-P. Schützenberger, “Polynômes de Schubert”, C. R. Acad. Sci. Paris Sér. I Math., 294:13 (1982), 447–450 | MR | Zbl

[12] E. Yu. Smirnov, A. A. Tutubalina, “Slide complexes and subword complexes”, Russian Math. Surveys, 75:6 (2020), 1162–1164 | DOI | DOI | MR | Zbl

[13] V. Pilaud, Ch. Stump, “EL-labelings and canonical spanning trees for subword complexes”, Discrete geometry and optimization, Fields Inst. Commun., 69, Springer, New York, 2013, 213–248 | DOI | MR | Zbl

[14] O. Pechenik, D. Searles, “Decompositions of Grothendieck polynomials”, Int. Math. Res. Not. IMRN, 2019:10 (2019), 3214–3241 | DOI | MR | Zbl