Equivariant Schubert calculus and jeu de taquin
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 275-318.

We introduce edge labeled Young tableaux. Our main results provide a corresponding analogue of Schützenberger’s theory of jeu de taquin. These are applied to the equivariant Schubert calculus of Grassmannians. Reinterpreting, we present new (semi)standard tableaux to study factorial Schur polynomials, after Biedenharn-Louck, Macdonald, Goulden-Greene, and others.

Consequently, we obtain new combinatorial rules for the Schubert structure coefficients, complementing work of Molev-Sagan, Knutson-Tao, Molev, and Kreiman. We also describe a conjectural generalization of one of our rules to the equivariant K-theory of Grassmannians, extending our previous work on non-equivariant K-theory. This conjecture concretely realizes the “positivity” known to exist by a result of Anderson-Griffeth-Miller. It provides an alternative to the conjectural rule of Knutson-Vakil.

Nous introduisons le concept de tableaux de Young avec arêtes étiquetées. Nos résultats principaux décrivent un analogue à la théorie du jeu de taquin de Schützenberger, avec applications au calcul de Schubert équivariant des grassmanniennes. Nous présentons de nouveaux tableaux (semi-)standards pour étudier les polynômes de Schur factoriels, d’après Biedenharn-Louck, Macdonald, et Goulden-Greene, entre autres.

Par conséquent, nous obtenons de nouvelles règles combinatoires pour les constantes de structure de Schubert, complémentaires aux travaux de Molev-Sagan, Knutson-Tao, Molev et Kreiman. Nous décrivons également une généralisation conjecturale d’une de nos règles à la K-théorie équivariante des grassmanniennes, étendant nos résultats précédents sur la K-théorie non équivariante. Cette conjecture réalise de façon concrète la positivité déjà connue par un résultat de Anderson-Griffeth-Miller, et offre une alternative à la règle conjecturale de Knutson-Vakil.

Received:
Accepted:
Published online:
DOI: 10.5802/aif.3161
Classification: 05E10,  14N15,  05E05,  57R91
Keywords: Schubert calculus, equivariant cohomology, Grassmannians, jeu de taquin
Thomas, Hugh 1; Yong, Alexander 2

1 Department of Mathematics and Statistics University of New Brunswick Fredericton, New Brunswick, E3B 5A3 (Canada)
2 Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801 (USA)
@article{AIF_2018__68_1_275_0,
     author = {Thomas, Hugh and Yong, Alexander},
     title = {Equivariant {Schubert} calculus and jeu de taquin},
     journal = {Annales de l'Institut Fourier},
     pages = {275--318},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3161},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3161/}
}
TY  - JOUR
TI  - Equivariant Schubert calculus and jeu de taquin
JO  - Annales de l'Institut Fourier
PY  - 2018
DA  - 2018///
SP  - 275
EP  - 318
VL  - 68
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3161/
UR  - https://doi.org/10.5802/aif.3161
DO  - 10.5802/aif.3161
LA  - en
ID  - AIF_2018__68_1_275_0
ER  - 
%0 Journal Article
%T Equivariant Schubert calculus and jeu de taquin
%J Annales de l'Institut Fourier
%D 2018
%P 275-318
%V 68
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3161
%R 10.5802/aif.3161
%G en
%F AIF_2018__68_1_275_0
Thomas, Hugh; Yong, Alexander. Equivariant Schubert calculus and jeu de taquin. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 275-318. doi : 10.5802/aif.3161. https://aif.centre-mersenne.org/articles/10.5802/aif.3161/

[1] Anderson, Dave; Griffeth, Stephen; Miller, Ezra Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces, J. Eur. Math. Soc. (JEMS), Volume 13 (2011) no. 1, pp. 57-84 | Article | Zbl: 1213.19003

[2] Biedenharn, Lawrence C.jun.; Louck, James D. A new class of symmetric polynomials defined in terms of tableaux, Adv. Appl. Math., Volume 10 (1989) no. 4, pp. 396-438 | Article | Zbl: 0698.05008

[3] Coskun, Izzet; Vakil, Ravi Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus, Algebraic Geometry (Seattle 2005, Pt. 1) (Proceedings of Symposia in Pure Mathematics) Volume 80, American Mathematical Society, Providence RI, 2009, pp. 77-124 | Zbl: 1184.14080

[4] Goulden, Ian; Greene, Curtis A new tableau representation for supersymmetric Schur functions, J. Algebra, Volume 170 (1994) no. 2, pp. 687-703 | Article | Zbl: 0840.20008

[5] Graham, William Positivity in equivariant Schubert calculus, Duke Math. J., Volume 109 (2001) no. 3, pp. 599-614 | Article | Zbl: 1069.14055

[6] Knutson, Allen Puzzles, positroid varieties, and equivariant K-theory of Grassmannians (2010) (https://arxiv.org/abs/1008.4302)

[7] Knutson, Allen; Miller, Ezra; Yong, Alexander Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math., Volume 630 (2009), pp. 1-31 | Article | Zbl: 1169.14033

[8] Knutson, Allen; Tao, Terence Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., Volume 119 (2003) no. 2, pp. 221-260 | Article | Zbl: 1064.14063

[9] Kreiman, Victor Equivariant Littlewood-Richardson skew tableaux, Trans. Am. Math. Soc., Volume 362 (2010) no. 5, pp. 2589-2617 | Article | Zbl: 1205.05244

[10] Macdonald, Ian Grant Schur functions: theme and variations, Sémin. Lothar. Comb., Volume 28 (1992), pp. 5-39 | Zbl: 0889.05073

[11] Manivel, Laurent Symmetric functions, Schubert polynomials and degeneracy loci, Texts and Monographs, Volume 6, American Mathematical Society, 2001, vii+167 pages | Zbl: 0998.14023

[12] Molev, Alexander I. Littlewood-Richardson polynomials, J. Algebra, Volume 321 (2009) no. 11, pp. 3450-3468 | Article | Zbl: 1169.05050

[13] Molev, Alexander I.; Sagan, Bruce E. A Littlewood-Richardson rule for factorial Schur functions, Trans. Am. Math. Soc., Volume 351 (1999) no. 11, pp. 4429-4443 | Article | Zbl: 0972.05053

[14] Pechenik, Olivier; Yong, Alexander Equivariant K-theory of Grassmannians, Forum of Mathematics, Pi, Volume 5 (2017) (https://doi.org/10.1017/fmp.2017.4) | Article

[15] Schützenberger, Marcel-Paul La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Strasbourg, 1976) (Lecture Notes in Math.) Volume 579, Springer, 1977, pp. 59-113 | Zbl: 0398.05011

[16] Thomas, Hugh; Yong, Alexander A combinatorial rule for (co)minuscule Schubert calculus, Adv. Math., Volume 222 (2009) no. 2, pp. 596-620 | Article | Zbl: 1208.14052

[17] Thomas, Hugh; Yong, Alexander A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus, Algebra Number Theory, Volume 3 (2009) no. 2, pp. 121-148 | Article | Zbl: 1229.05285

[18] Woo, Alexander; Yong, Alexander A Gröbner basis for Kazhdan-Lusztig ideals, Am. J. Math., Volume 134 (2012) no. 4, pp. 1089-1137 | Article | Zbl: 1262.13044

[19] Zinn-Justin, Paul Littlewood-Richardson coefficients and integrable tilings, Electron. J. Comb., Volume 16 (2009) no. 1 (Research Paper R12, 33 p.) | Zbl: 1184.05133

Cited by Sources: