Categories for Grassmannian Cluster Algebras of Infinite Rank

Author:

August Jenny1,Cheung Man-Wai2,Faber Eleonore3,Gratz Sira1,Schroll Sibylle45

Affiliation:

1. Department of Mathematics , Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark

2. School of Mathematics , Kavli IPMU (WPI), UTIAS, The University of Tokyo, Japan

3. School of Mathematics , University of Leeds, Leeds, LS2 9JT, UK

4. Department of Mathematics , Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

5. Institutt for matematiske fag , NTNU, N-7491 Trondheim, Norway

Abstract

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 $\textrm {Ext}^{1}$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.

Funder

Max Planck Institute for Mathematics

Danish National Research Foundation

NSF

AMS Simons Travel Grants

Marie Skłodowska-Curie fellow at the University of Leeds

EPSRC

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference25 articles.

1. On a category of cluster algebras;Assem;J. Pure Appl. Algebra,2014

2. Cluster structures for the ${A}_{\infty }$ singularity;August,2022

3. Dimer models and cluster categories of Grassmannians;Baur;Proc. London Math. Soc. (3),2016

4. Maximal Cohen–Macaulay Modules and Tate Cohomology

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Cluster structures for the A∞$A_\infty$ singularity;Journal of the London Mathematical Society;2023-04-15

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