Projected Richardson varieties and affine Schubert varieties

He, Xuhua; Lam, Thomas

doi:10.5802/aif.2990

Projected Richardson varieties and affine Schubert varieties
[Variétés de Richardson projetées et variétés de Schubert affines]

He, Xuhua ¹ ; Lam, Thomas ²

¹ Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay Kowloon, Hong Kong (China)
² Department of Mathematics University of Michigan Ann Arbor MI 48109 (USA)

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2385-2412.

Résumé
Abstract

Soit $G$ un groupe algébrique complexe quasi-simple et $G / P$ une variété de drapeaux partiels. La projection sur $G / P$ des variétés de Richardson (de la variété des drapeaux complets) forment une stratification de $G / P$ . Nous montrons que les relations d’adhérence des variétés de Richardson projetées correspondent à celles d’un certain sous-ensemble de variétés de Schubert sur la variété de drapeaux affine de $G$ . Nous comparons aussi les classes de cohomologie équivariante et de $K$ -théorie de ces deux stratifications. Notre travail généralise celui de Knutson, Lam et Speyer pour la grassmannienne de type $A$ .

Let $G$ be a complex quasi-simple algebraic group and $G / P$ be a partial flag variety. The projections of Richardson varieties from the full flag variety form a stratification of $G / P$ . We show that the closure partial order of projected Richardson varieties agrees with that of a subset of Schubert varieties in the affine flag variety of $G$ . Furthermore, we compare the torus-equivariant cohomology and $K$ -theory classes of these two stratifications by pushing or pulling these classes to the affine Grassmannian. Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian of type $A$ .

DOI : 10.5802/aif.2990

Classification : 14M15, 05E10
Keywords: flag variety, Schubert calculus, projected Richardson variety, affine Schubert variety
Mot clés : Variété de drapeaux, calcul de Schubert, variété de Richardson projetée, variété de Schubert affine

Affiliations des auteurs :

He, Xuhua ¹ ; Lam, Thomas ²

¹ Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay Kowloon, Hong Kong (China)
² Department of Mathematics University of Michigan Ann Arbor MI 48109 (USA)

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     title = {Projected {Richardson} varieties and affine {Schubert} varieties},
     journal = {Annales de l'Institut Fourier},
     pages = {2385--2412},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     doi = {10.5802/aif.2990},
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He, Xuhua; Lam, Thomas. Projected Richardson varieties and affine Schubert varieties. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2385-2412. doi : 10.5802/aif.2990. https://aif.centre-mersenne.org/articles/10.5802/aif.2990/

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