The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

Muthiah, D. , Weekes, A. and Yacobi, O. (2020) The equations defining affine Grassmannians in type A and a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman. International Mathematics Research Notices, 2022(3), pp. 1922-1972. (doi: 10.1093/imrn/rnaa131)

Abstract

The affine Grassmannian of SLₙ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called shuffles, and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps; first, we reinterpret the shuffle equations in terms of Frobenius twists of symmetric functions. Using this, we reduce to a finite-dimensional problem, which we solve. For the 2nd step, we introduce a finite-dimensional analogue of the affine Grassmannian of SLₙ⁠, which we conjecture to be precisely the reduced subscheme of a finite-dimensional Grassmannian consisting of subspaces invariant under a nilpotent operator.

• Item Type: Articles
• Additional Information: D.M. was supported by a PIMS Postdoctoral Fellowship and by JSPS KAKENHI grant number JP19K14495. A.W. was supported in part by a Government of Ontario graduate scholarship. O.Y. is supported by the Australian Research Council Discovery Project grant DP180102563. This research is supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Muthiah, Dr Dinakar
• Authors: Muthiah, D., Weekes, A., and Yacobi, O.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: International Mathematics Research Notices
• Publisher: Oxford University Press
• ISSN: 1073-7928
• ISSN (Online): 1687-0247