Grassmannians and Cluster Algebras

TL;DR

This paper establishes that the homogeneous coordinate ring of Grassmannians forms a cluster algebra of geometric type, classifies those with finite cluster type, and explores the geometric significance of cluster variables.

Contribution

It proves the cluster algebra structure of Grassmannian coordinate rings and classifies finite cluster types, linking algebraic and geometric properties.

Findings

Grassmannian coordinate rings are cluster algebras of geometric type

Finite cluster types of Grassmannians are classified

Cluster variables relate to configurations of points in projective space

Abstract

This paper demonstrates that the homogeneous coordinate ring of the Grassmannian $\Bbb{G}(k,n)$ is a {\it cluster algebra of geometric type} - as defined by S. Fomin and A. Zelevinsky. Grassmannians having {\it finite cluster type} are classified and the associated cluster variables are studied in connection with the geometry of configurations of points in $\Bbb{R}\Bbb{P}^2$.