Connections to Schubert Calculus Learning Seminar

There is a bijection between solutions of the Kadomtsev-Petiashvili (KP) hierarchy and points on an infinite Grassmannian, now often simply referred to as "the Sato Grassmannian". This connection is made via the Plücker coordinates, the expansion coefficients of the KP tau-function in the basis of Schur functions. Sato's result is that the tau-functions satisfy the Plücker relations of all finite Grassmannians and their union forms what is known as the KP hierarchy. In other words, one considers the inductive limit of the coordinate rings of finite Grassmannians. A celebrated result is that the coordinate rings of finite Grassmannians carry a cluster algebra structure. We show that this cluster algebra structure can be extended to the inductive limit by introducing ind-cluster algebras. This ind-cluster algebra structure allows us to address algebraic independence of Plücker relations and positivity. This is joint work with Sira Gratz, Aarhus University.