Grassmann–Grassmann conormal varieties, integrability, and plane partitions
Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1087-1145.

We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant K-class is given by the partition function of an integrable loop model, and furthermore their K-theoretic pushforward to a point is a solution of the level 1 quantum Knizhnik–Zamolodchikov equation. We prove these results in the case that the Lagrangian is smooth (hence is the conormal bundle to a subGrassmannian). To compute the pushforward to a point, or equivalently to the affinization, we simultaneously degenerate the Lagrangian and sheaf (over the affinization); the sheaf degenerates to a direct sum of cyclic modules over the geometric components, which are in bijection with plane partitions, giving a geometric interpretation to the Razumov–Stroganov correspondence satisfied by the loop model.

Nous donnons une formule conjecturelle pour des faisceaux coherents de support des variétés conormales dans le fibré cotangent de la Grassmannienne, tels que leur classe de K-théorie équivariante est donnée par la fonction de partition d’un modèle de boucles intégrable, et que de plus leur image dans la K-théorie d’un point est solution de l’équation de Knizhnik–Zamolodchikov quantique de niveau 1. Nous démontrons ces résultats dans le cas où la Lagrangienne est lisse (donc le fibré conormal d’une sous-Grassmannienne). Pour pousser en avant vers un point, ou de manière équivalente vers son affinisation, nous dégénérons simultanément la Lagrangienne et son faisceau (sur l’affinisation) ; le faisceau dégénère en une somme directe de modules cycliques sur les composantes géométriques, qui sont en bijection avec des partitions planes, ce qui donne une interprétation géométrique à la correspondance de Razumov–Stroganov satisfaite par le modèle de boucles.

Published online:
DOI: 10.5802/aif.3266
Classification: 14M15, 82B23, 14Q99
Keywords: Quantum Knizhnik–Zamolodchikov equation, equivariant $K$-theory, cotangent bundle of the Grassmannian, loop model
Knutson, Allen 1; Zinn-Justin, Paul 2

1 Cornell Ithaca, NY 14853 (USA)
2 School of Mathematics and Statistics The University of Melbourne Parkville, Victoria 3010 (Australia)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Knutson, Allen; Zinn-Justin, Paul. Grassmann–Grassmann conormal varieties, integrability, and plane partitions. Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1087-1145. doi : 10.5802/aif.3266.

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