College of Science and Mathematics
Department of Mathematics
Mathematics Seminar Series - Fall 2002
Monday,
December 2, 2002
2:30 pm, Science 2-065
2:30 pm, Science 2-065
Joshua Scott
Northeastern University andMassachusetts Institute of Technology
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 $k=3 , 8 \geq n \geq 5$.
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 $k=3 , 8 \geq n \geq 5$.
The presentations cover a large variety of topics and are intended for a general math audience. The seminar is organized by Prof. Alfred Noël and we usually meet Monday afternoons, from 2:30 pm to 4:00 pm.
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