An equivariant rim hook rule for quantum cohomology of Grassmannians

Abstract

International audience 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.

Equivariant quantum cohomology and Yang-Baxter algebras

“…During the writing up of this manuscript two works appeared on different combinatorial approaches to compute Gromov-Witten invariants. The work [5] proves a conjecture of Knutson which states that puzzles for two-step flag varieties describe the product of non-equivariant quantum cohomology for Grassmannians, while the work [2] describes a generalised rim-hook formula to compute equivariant Gromov-Witten invariants for Grassmannians. We hope to address how these latter combinatorial approaches are related to formula (6.32) in future work.…”

“…, corresponding to the toric tableau and the associated integers ℓ appearing in the second product of (4.8) are detailed in the table below, For instance, we find for i = 1, 2, 3 and j = 2 that λ (2) [d 2 ] + ρ + 1 = (. .…”

“…C. K. gratefully acknowledges discussions with Gwyn Bellamy and V. G. would like to thank the Max Planck Institute for Mathematics Bonn where part of this work was carried out. Further thanks are due to Catharina Stroppel for her detailed and helpful comments on a draft manuscript, Jessica Striker for alerting us to the work [2] and Kaisa Taipale for making an extended abstract available.…”

Quantum integrability and generalised quantum Schubert calculus

“…Our proof of the last theorem contains an algorithm for the successive computation of the structure constants C ν λµ (t, q) without making use of the explicit solutions of the Bethe ansatz equations (4.17) and the residue formula (4.36). Namely, starting from the Pieri rule (2.22) for G 1 , one can use (4.21) and (2.17) to define a generalised version of the rim-hook algorithm at β = 0 [7]; see [5] for a recent extension to the equivariant case with β = 0. We shall demonstrate this only on a simple example.…”

Yang-Baxter algebras, convolution algebras, and Grassmannians

“…The discussion with the D-operator can be found in [Kor14, Lemmas 3.12, 3.14 and 3.20]. The equivariant case of the rim-hook algorithm has been discussed more recently in [BBT14]. The formulation using (algebraic) D-operators appears already in [GoKo17], which also covers the case of equivariant quantum K-theory.…”