Monday, December 2nd, 2002
2:30pm - 3:30pm, in Science 2-065

Joshua Scott

Northeastern University and MIT

Cluster Algebras and Grassmannians

Abstract: In 2000 S. Fomin and A. Zelevinsky introduced the theory of Cluster Algebras as an apparatus to study the structure of dual canonical bases for algebraic groups. This robust field has since found manifold applications in the study of totally positive matrices, non-linear recurrence relations, Poisson geometry, and quantum groups. In its most naive incarnation, a cluster algebra is a commutative ring generated inside an ambient field by a family of distinguished generators called cluster variables which are grouped into families or clusters. These generators are produced, recursively, inside the ambient field by means of a process called mutation from an initial fixed family of indeterminates and a fixed skew-symmetric matrix. I will show that the homogeneous coordinate algebra of the Grassmannian $\Bbb{C} \Big[ \Bbb{G}(k,n) \Big]$ is a {\it Cluster Algebra} whose ambient field is simply the field of rational functions $\Bbb{C} \Big( \Bbb{G}(k,n) \Big)$ and whose initial cluster consists entirely of special Plücker coordinates. My talk will highlight a combinatorial device, introduced by my colleague Alexander Postnikov, which is instrumental in determining this initial family of Plücker coordinates. If time permits, I will exhibit a new basis for $\Bbb{C} \Big[ \Bbb{G}(k,n) \Big]$ arising from the cluster algebra structure in the cases $k=2 , n \geq 4$ and $k=3 , 8 \geq n \geq 5$.