Zero-one Schubert polynomials. (English) Zbl 1464.14057

Fink, Alex; Mészáros, Karola; Dizier, Avery St.

Math. Z. 297, No. 3-4, 1023-1042 (2021).

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.

Reviewer: Evgeny Smirnov (Moskva)

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

Citations:

Zbl 1443.05179

References:

[1]	Bergeron, N.; Billey, S., RC-graphs and Schubert polynomials, Exp. Math., 2, 4, 257-269 (1993) · Zbl 0803.05054 · doi:10.1080/10586458.1993.10504567
[2]	Billey, S.; Jockusch, W.; Stanley, RP, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., 2, 4, 345-374 (1993) · Zbl 0790.05093 · doi:10.1023/A:1022419800503
[3]	Demazure, M., Une nouvelle formule des caracteres, J. Combin. Theory Ser. A, 70, 1, 107-143 (1995) · Zbl 0819.05058 · doi:10.1016/0097-3165(95)90083-7
[4]	Fink, A., Mészáros, K., St, A., Dizier, : Schubert polynomials as integer point transforms of generalized permutahedra. Adv. Math. 332, 465-475 (2018) · Zbl 1443.05179
[5]	Fomin, S.; Kirillov, AN, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math., 153, 1, 123-143 (1996) · Zbl 0852.05078 · doi:10.1016/0012-365X(95)00132-G
[6]	Fomin, S.; Stanley, RP, Schubert polynomials and the nilCoxeter algebra, Adv. Math., 103, 2, 196-207 (1994) · Zbl 0809.05091 · doi:10.1006/aima.1994.1009
[7]	Knutson, A.; Miller, E., Gröbner geometry of Schubert polynomials, Ann. Math. (2), 161, 3, 1245-1318 (2005) · Zbl 1089.14007 · doi:10.4007/annals.2005.161.1245
[8]	Kraśkiewicz, W.; Pragacz, P., Foncteurs de Schubert, C. R. Acad. Sci. Paris Sér. I Math., 304, 9, 209-211 (1987) · Zbl 0642.13011
[9]	Lam, T., Lee, S., Shimozono, M.: Back stable Schubert calculus, Jun 2018. arxiv: 1806.11233
[10]	Lascoux, A.; Schützenberger, M-P, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér I Math, 294, 13, 447-450 (1982) · Zbl 0495.14031
[11]	Lascoux, A.; Schützenberger, M-P, Keys and standard bases, IMA Math. Appl., 19, 125-144 (1990) · Zbl 0815.20013
[12]	Lenart, C., A unified approach to combinatorial formulas for Schubert polynomials, J. Algebraic Combin., 20, 3, 263-299 (2004) · Zbl 1056.05146 · doi:10.1023/B:JACO.0000048515.00922.47
[13]	Magyar, P., Schubert polynomials and Bott-Samelson varieties, Comment. Math. Helv., 73, 4, 603-636 (1998) · Zbl 0951.14036 · doi:10.1007/s000140050071
[14]	Manivel, L.: Symmetric functions, Schubert polynomials and degeneracy loci, volume 6 of SMF/AMS Texts and Monographs. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow, Cours Spécialisés [Specialized Courses], 3 · Zbl 0998.14023
[15]	Monical, C., Tokcan, N., Yong, A.: Newton polytopes in algebraic ombinatorics. Selecta Math. (N.S.) 25, 66 (2019) · Zbl 1426.05175
[16]	Reiner, V.; Shimozono, M., Key polynomials and a flagged Littlewood-Richardson rule, J. Combin. Theory Ser. A, 70, 1, 107-143 (1995) · Zbl 0819.05058 · doi:10.1016/0097-3165(95)90083-7
[17]	Weigandt, A.; Yong, A., The prism tableau model for Schubert polynomials, J. Comb. Theory Ser. A, 154, 551-582 (2018) · Zbl 1373.05219 · doi:10.1016/j.jcta.2017.09.009