Categories for Grassmannian cluster algebras of infinite rank

TL;DR

This paper constructs infinite-rank Grassmannian categories linked to cluster algebras, establishing a bijection between certain modules and Plücker coordinates, and develops formulas for Ext space dimensions.

Contribution

It introduces a new class of infinite-rank Grassmannian categories and relates their modules to infinite Grassmannian cluster algebra structures.

Findings

Bijective correspondence between rank 1 modules and Plücker coordinates

Structure-preserving relation between rigidity and compatibility

Explicit combinatorial formula for Ext^1 space dimensions

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 Pl\"ucker 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 Pl\"ucker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\mathrm{Ext}^1$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.