Joshua Scott

Abstract:   This dissertation is comprised of two related investigations. The first addresses properties of the quantum Grassmannian while the second studies algebraic and combinatorial features of the Grassmannian's classical homogeneous coordinate ring C[G(k,n)]. The primary focus of the defense will be on the latter.

Following the program developed by Sergey Fomin and Andrei Zelevinsky, this dissertation demonstrates that C[G(k,n)] is a cluster algebra of geometric type. The proof highlights a generalization of pseudo-line arrangements, introduced by Alexander Postnikov, and is used to manufacture clusters consisting entirely of Plücker coordinates. Special attention is given to those Grassmannians possessing finitely many clusters; in particular cluster variables attached to these Grassmannians are studied in connection with the geometry of configurations of points in CP².