Geodesic
Inclusion-exclusion on Schubert polynomials
Mészáros, Karola1 ; Tanjaya, Arthur2
1 Department of Mathematics Cornell University Ithaca, NY 14853 USA.
2 Department of Mathematics Cornell University Ithaca NY 14853 USA.
Algebraic Combinatorics, Tome 5 (2022) no. 2, pp. 209-226.

Voir la notice de l'article provenant de la source Numdam

We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns 1432 and 1423 is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about principal specializations of Schubert polynomials. We propose a general framework for finding inclusion-exclusion inspired expression of Schubert polynomials of all permutations.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/alco.200
Classification : 05E05
Keywords: Schubert polynomial, principal specialization, nonnegative linear combination.

Mészáros, Karola 1 ; Tanjaya, Arthur 2

1 Department of Mathematics Cornell University Ithaca, NY 14853 USA.
2 Department of Mathematics Cornell University Ithaca NY 14853 USA.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{ALCO_2022__5_2_209_0,
     author = {M\'esz\'aros, Karola and Tanjaya, Arthur},
     title = {Inclusion-exclusion on {Schubert} polynomials},
     journal = {Algebraic Combinatorics},
     pages = {209--226},
     publisher = {The Combinatorics Consortium},
     volume = {5},
     number = {2},
     year = {2022},
     doi = {10.5802/alco.200},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5802/alco.200/}
}
TY  - JOUR
AU  - Mészáros, Karola
AU  - Tanjaya, Arthur
TI  - Inclusion-exclusion on Schubert polynomials
JO  - Algebraic Combinatorics
PY  - 2022
SP  - 209
EP  - 226
VL  - 5
IS  - 2
PB  - The Combinatorics Consortium
UR  - http://geodesic.mathdoc.fr/articles/10.5802/alco.200/
DO  - 10.5802/alco.200
LA  - en
ID  - ALCO_2022__5_2_209_0
ER  - 
%0 Journal Article
%A Mészáros, Karola
%A Tanjaya, Arthur
%T Inclusion-exclusion on Schubert polynomials
%J Algebraic Combinatorics
%D 2022
%P 209-226
%V 5
%N 2
%I The Combinatorics Consortium
%U http://geodesic.mathdoc.fr/articles/10.5802/alco.200/
%R 10.5802/alco.200
%G en
%F ALCO_2022__5_2_209_0
Mészáros, Karola; Tanjaya, Arthur. Inclusion-exclusion on Schubert polynomials. Algebraic Combinatorics, Tome 5 (2022) no. 2, pp. 209-226. doi : 10.5802/alco.200. http://geodesic.mathdoc.fr/articles/10.5802/alco.200/

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

[2] Billey, Sara C.; Holroyd, Alexander E.; Young, Benjamin J. A bijective proof of Macdonald’s reduced word formula, Algebr. Comb., Volume 2 (2019) no. 2, pp. 217-248 | mathdoc-id | Zbl | MR | DOI

[3] Billey, Sara C.; Jockusch, William; Stanley, Richard P. Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., Volume 2 (1993) no. 4, pp. 345-374 | Zbl | MR | DOI

[4] Fan, Neil J.; Guo, Peter L. Upper Bounds of Schubert Polynomials, Sci. China Math. (2021) | DOI

[5] Fink, Alex; Mészáros, Karola; St. Dizier, Avery Schubert polynomials as integer point transforms of generalized permutahedra, Adv. Math., Volume 332 (2018), pp. 465-475 | Zbl | MR | DOI

[6] Fink, Alex; Mészáros, Karola; St. Dizier, Avery Zero-one Schubert polynomials, Math. Z., Volume 297 (2021) no. 3-4, pp. 1023-1042 | Zbl | MR | DOI

[7] Fomin, Sergey; Kirillov, Anatol N. The Yang–Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math., Volume 153 (1996) no. 1-3, pp. 123-143 Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993) | MR | Zbl | DOI

[8] Fomin, Sergey; Kirillov, Anatol N. Reduced words and plane partitions, J. Algebraic Combin., Volume 6 (1997) no. 4, pp. 311-319 | Zbl | MR | DOI

[9] Fomin, Sergey; Stanley, Richard P. Schubert polynomials and the nilCoxeter algebra, Adv. Math., Volume 103 (1994) no. 2, pp. 196-207 | Zbl | MR | DOI

[10] Gao, Yibo Principal specializations of Schubert polynomials and pattern containment, European J. Combin., Volume 94 (2021), 103291, 12 pages | Zbl | MR | DOI

[11] Huh, June; Matherne, Jacob P.; Mészáros, Karola; St. Dizier, Avery Logarithmic concavity of Schur and related polynomials, Trans. Am. Math. Soc. (2022) (https://doi.org/10.1090/tran/8606)

[12] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1245-1318 | Zbl | MR | DOI

[13] Kraśkiewicz, Witold; Pragacz, Piotr Foncteurs de Schubert, C. R. Acad. Sci. Paris Sér. I Math., Volume 304 (1987) no. 9, pp. 209-211 | Zbl | MR

[14] Lam, Thomas; Lee, Seung Jin; Shimozono, Mark Back stable Schubert calculus, Compos. Math., Volume 157 (2021) no. 5, pp. 883-962 | Zbl | MR | DOI

[15] Lascoux, Alain; Schützenberger, Marcel-Paul Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math., Volume 294 (1982) no. 13, pp. 447-450 | Zbl | MR | DOI

[16] Lenart, Cristian A unified approach to combinatorial formulas for Schubert polynomials, J. Algebraic Combin., Volume 20 (2004) no. 3, pp. 263-299 | Zbl | MR | DOI

[17] Macdonald, Ian G. Notes on Schubert polynomials, Publ. LaCIM, UQAM, Montréal, 1991

[18] Magyar, Peter Schubert polynomials and Bott–Samelson varieties, Comment. Math. Helv., Volume 73 (1998) no. 4, pp. 603-636 | Zbl | MR | DOI

[19] Manivel, Laurent Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, 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) | Zbl | MR

[20] Monical, Cara; Tokcan, Neriman; Yong, Alexander Newton polytopes in algebraic combinatorics, Selecta Math. (N.S.), Volume 25 (2019) no. 5, 66, 37 pages | Zbl | MR | DOI

[21] Morales, Alejandro H.; Pak, Igor; Panova, Greta Asymptotics of principal evaluations of Schubert polynomials for layered permutations, Proc. Amer. Math. Soc., Volume 147 (2019) no. 4, pp. 1377-1389 | Zbl | MR | DOI

[22] Reiner, Victor; Shimozono, Mark Key polynomials and a flagged Littlewood–Richardson rule, J. Combin. Theory Ser. A, Volume 70 (1995) no. 1, pp. 107-143 | Zbl | MR | DOI

[23] Stanley, Richard P. Some Schubert shenanigans (2017) (https://arxiv.org/abs/1704.00851)

[24] Weigandt, Anna; Yong, Alexander The prism tableau model for Schubert polynomials, J. Combin. Theory Ser. A, Volume 154 (2018), pp. 551-582 | Zbl | MR | DOI

[25] Weigandt, Anna E. Schubert polynomials, 132-patterns, and Stanley’s conjecture, Algebr. Comb., Volume 1 (2018) no. 4, pp. 415-423 | mathdoc-id | Zbl | MR | DOI

Cité par Sources :