2025/05/02 by Gratz, Sira, Korff, Christian
#05E05 #37K10 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 13F60 #Representation Theory (math.RT) #Secondary: 14M15
paper · doi:10.48550/arxiv.2505.01228
A prototypical examples of a cluster algebra is the coordinate ring of a finite Grassmannian: using the Plücker embedding the cluster algebra structure allows one to move between `maximal sets' of algebraically independent Plücker 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.