Equivariant Schubert calculus and jeu de taquin

Thomas, Hugh; Yong, Alexander

Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 275-318.

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.

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.

Classification : 05E10, 14N15, 05E05, 57R91
Keywords: Schubert calculus, equivariant cohomology, Grassmannians, jeu de taquin
Mot clés : Calcul de Schubert, cohomologie équivariante, grassmanniennes, jeu de taquin

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