Grassmann-Grassmann conormal varieties, integrability, and plane partitions. (Variétés conormales Grassmann-Grassmann, intégrabilité et partitions planes.) (English. French summary) Zbl 1471.14104
Cohomological or \(K\)-theoretic classes of Schubert subvarieties of Grassmannians play an important role in enumerative geometry and representation theory. Recent works indicate the importance of replacing the Grassmannian with the total space of its cotangent bundle. The natural counterpart of a Schubert variety in this setting is the conormal bundle of a Schubert variety.
The present paper considers the classes of conormal bundles of Schubert varieties in equivariant \(K\)-theory. Moreover, and very importantly, not only the K theory class of the conormal bundle, but the representing sheave as well. The authors present a conjectured formula for that sheave, as a rectangular domain partition function. They verify the conjecture when the Schubert variety is smooth.
The authors compute that the \(K\)-theory class of their sheave is a partition function of an integrable loop model. Also, when pushed forward to a point, it is a solution to the level-1 Knizhnik-Zamolodchikov equation. The computation of the push-forward to a point is carried out by a simultaneous degeneration of the conormal bundle and of the sheave. The sheave degeneration turns out to follow the combinatorics of analogous processes for plane partitions. As result, the authors obtain a geometric interpretation of the Razumov-Stroganov correspondence.
The present paper considers the classes of conormal bundles of Schubert varieties in equivariant \(K\)-theory. Moreover, and very importantly, not only the K theory class of the conormal bundle, but the representing sheave as well. The authors present a conjectured formula for that sheave, as a rectangular domain partition function. They verify the conjecture when the Schubert variety is smooth.
The authors compute that the \(K\)-theory class of their sheave is a partition function of an integrable loop model. Also, when pushed forward to a point, it is a solution to the level-1 Knizhnik-Zamolodchikov equation. The computation of the push-forward to a point is carried out by a simultaneous degeneration of the conormal bundle and of the sheave. The sheave degeneration turns out to follow the combinatorics of analogous processes for plane partitions. As result, the authors obtain a geometric interpretation of the Razumov-Stroganov correspondence.
Reviewer: Richárd Rimányi (Chapel Hill)
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |
| 82B23 | Exactly solvable models; Bethe ansatz |
Keywords:
quantum Knizhnik-Zamolodchikov equation; equivariant \(K\)-theory; cotangent bundle of the Grassmannian; loop modelSoftware:
Macaulay2References:
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