Categories for Grassmannian cluster algebras of infinite rank

August, J. , Cheung, M.-W., Faber, E., Gratz, S. and Schroll, S. (2024) Categories for Grassmannian cluster algebras of infinite rank. International Mathematics Research Notices, 2024(2), pp. 1166-1210. (doi: 10.1093/imrn/rnad004)

Abstract

We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen–Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank 1 in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the Ext1-spaces between any two generically free modules of rank 1 in the Grassmannian category of infinite rank.

• Item Type: Articles
• Additional Information: This work was supported by the Max Planck Institute for Mathematics; the Danish National Research Foundation [DNRF156 to J.A.] the NSF [DMS-1854512 to M.C.]; the AMS Simons Travel Grants; a Marie Skłodowska-Curie fellowship at the University of Leeds [789580 to E.F.]; and the EPSRC [EP/W007509/1 to E.F. and EP/P016294/1 to S.S.].
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: August, Dr Jenny
• Authors: August, J., Cheung, M.-W., Faber, E., Gratz, S., and Schroll, S.
• College/School: University Services > Learning and Teaching Services Division
• Journal Name: International Mathematics Research Notices
• Publisher: Oxford University Press
• ISSN: 1073-7928
• ISSN (Online): 1687-0247
• Published Online: 27 February 2023
• Copyright Holders: Copyright © The Author(s) 2023
• First Published: First published in International Mathematics Research Notices 2024(2):1166–1210
• Publisher Policy: Reproduced in accordance with the publisher copyright policy