Abstract
A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the quantum product in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of $k$-planes in complex $n$-space by doing classical multiplication and then applying a combinatorial rimhook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule provides an effective algorithm for computing the equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus. Une question importante dans la cohomologie quantique des variétés de drapeaux est de trouver des formules positives non récursives pour exprimer le produit quantique dans une base particulièrement bonne, appelée la base de Schubert. Bertram, Ciocan-Fontanine et Fulton donnent une façon de calculer les produits quantiques de classes de Schubert dans la Grassmannienne de $k$-plans dans l’espace complexe de dimension $n$ en faisant la multiplication classique et appliquant une règle combinatoire “rimhook” qui donne le paramètre quantique. Dans cet article, nous donnons une généralisation de ce règle rimhook au contexte où il y a aussi une action du tore complexe. Combiné avec la règle “puzzle” de Knutson et Tao, cela donne une algorithme effective pour calculer les coefficients équivariants de Littlewood-Richard. Il est intéressant d'observer que cette règle demande une spécialisation des poids du tore qui est similaire d’une manière tentante aux applications dans le calcul de Schubert affiné.
Highlights
Quantum cohomology grew out of explorations in string theory in the early 1990s
The proposal by Candelas, de la Ossa, Green, and Parkes of a partial answer to the Clemens conjecture regarding the number of rational curves of given degree on a general quintic threefold brought enormous attention to the mathematical ideas being used by these physicists
This project extends some of the beautiful tools of combinatorics to an easier method for computing equivariant quantum Littlewood-Richardson (EQLR) coefficients. (Small) quantum cohomology is a deformation of classical cohomology by the quantum parameter q, and the Schubert basis elements σλ in H∗(Gr(k, n)) miraculously still form a basis for both the quantum and equivariant quantum cohomology of the Grassmannian
Summary
Quantum cohomology grew out of explorations in string theory in the early 1990s. The proposal by Candelas, de la Ossa, Green, and Parkes of a partial answer to the Clemens conjecture regarding the number of rational curves of given degree on a general quintic threefold brought enormous attention to the mathematical ideas being used by these physicists. The reduction of torus weights modulo n in the main theorem of this paper appears in the context of Lam and Shimozono’s work relating double quantum Schubert polynomials to k-double Schur polynomials [LS11]. It is the expectation of the authors that cyclic factorial Schur polynomials are the image of the k-double Schur polynomials, which are known to represent equivariant homology classes of the affine Grassmannian, under the Peterson isomorphism [LS13]. This connection suggests that the equivariant rim hook rule is a shadow of Peterson’s isomorphism and can shed further light on what has become known as the “quantum equals affine” phenomenon, and the authors intend to explore this in a future paper
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