Ind-cluster algebras and infinite Grassmannians

TL;DR

This paper demonstrates that the coordinate ring of the infinite Grassmannian can be structured as an infinite rank cluster algebra, linking finite Grassmannian cluster structures with the infinite case via colimit constructions.

Contribution

It establishes that cluster algebras of infinite rank are exactly the ind-objects of a natural category of cluster algebras, connecting finite and infinite Grassmannian structures.

Findings

The coordinate ring of the infinite Grassmannian is a cluster algebra of infinite rank.

Cluster algebras of infinite rank are characterized as ind-objects of a natural category.

The structure of the infinite Grassmannian cluster algebra is induced by colimit construction.

Abstract

A prototypical examples of a cluster algebra is the coordinate ring of a finite Grassmannian: using the Pl\"ucker embedding the cluster algebra structure allows one to move between `maximal sets' of algebraically independent Pl\"ucker coordinates via mutations. Fioresi and Hacon studied a specific colimit of the coordinate rings of finite Grassmannians and its link with the infinite Grassmannian introduced by Sato and independently by Segal and Wilson in connection with the Kadomtsev-Petiashvili (KP) hierarchy, an infinite set of nonlinear partial differential equations which possess soliton solutions. In this article we prove that this ring is a cluster algebra of infinite rank with the structure induced by the colimit construction. More generally, we prove that cluster algebras of infinite rank are precisely the ind-objects of a natural category of cluster algebras.