ALGEBRAIC COMBINATORICS

no. 3
An involution on RC-graphs and a conjecture on dual Schubert polynomials by Postnikov and Stanley
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 593-602.

In this paper, we provide explicit formula for the dual Schubert polynomials of a special class of permutations using certain involution principals on RC-graphs, resolving a conjecture by Postnikov and Stanley.

Received: 2019-01-02
Revised: 2019-11-29
Accepted: 2019-12-15
Published online: 2020-06-02
DOI: https://doi.org/10.5802/alco.105
Classification: 05E05,  14N15
Keywords: Schubert polynomial, dual Schubert polynomial, Bruhat chains 
@article{ALCO_2020__3_3_593_0,
     author = {Gao, Yibo},
     title = {An involution on RC-graphs and a conjecture on dual Schubert polynomials by Postnikov and Stanley},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {3},
     year = {2020},
     pages = {593-602},
     doi = {10.5802/alco.105},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_3_593_0/}
}
Gao, Yibo. An involution on RC-graphs and a conjecture on dual Schubert polynomials by Postnikov and Stanley. Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 593-602. doi : 10.5802/alco.105. https://alco.centre-mersenne.org/item/ALCO_2020__3_3_593_0/

[1] Bergeron, Nantel; Billey, Sara C. RC-graphs and Schubert polynomials, Exp. Math., Volume 2 (1993) no. 4, pp. 257-269 | Article | MR 1281474 | Zbl 0803.05054

[2] Billey, Sara C.; Jockusch, William; Stanley, Richard P. Some combinatorial properties of Schubert polynomials, J. Algebr. Comb., Volume 2 (1993) no. 4, pp. 345-374 | Article | MR 1241505 | Zbl 0790.05093

[3] Fomin, Sergey; Kirillov, Anatol N. Yang–Baxter equation, symmetric functions and Grothendieck polynomials (1993) (arXiv preprint https://arxiv.org/abs/hep-th/9306005)

[4] Fomin, Sergey; Stanley, Richard P. Schubert polynomials and the nil-Coxeter algebra, Adv. Math., Volume 103 (1994) no. 2, pp. 196-207 | Article | MR 1265793 | Zbl 0809.05091

[5] Gaetz, Christian; Gao, Yibo Padded Schubert polynomials and weighted enumeration of Bruhat chains (2019) (arXiv preprint https://arxiv.org/abs/1905.00047)

[6] Gaetz, Christian; Gao, Yibo A combinatorial duality between the weak and strong Bruhat orders, J. Comb. Theory, Ser. A, Volume 171 (2020), pp. 105-178 | Article | MR 4040744 | Zbl 07172386

[7] Manivel, Laurent Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, Volume 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001, viii+167 pages (Translated from the 1998 French original by John R. Swallow, Cours Spécialisés [Specialized Courses], 3) | MR 1852463 | Zbl 0998.14023

[8] Postnikov, Alexander; Stanley, Richard P. Chains in the Bruhat order, J. Algebr. Comb., Volume 29 (2009) no. 2, pp. 133-174 | Article | MR 2475632 | Zbl 1238.14036

[9] Stembridge, John R. A weighted enumeration of maximal chains in the Bruhat order, J. Algebr. Comb., Volume 15 (2002) no. 3, pp. 291-301 | Article | MR 1900629 | Zbl 1008.20037