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Cyclic polytopes, analogues of tropical cluster algebras, and higher Auslander algebrasAlgebra & Discrete Mathematics
|Speaker: ||Hugh Thomas, University of New Brunswick|
|Location: ||2112 MSB|
|Start time: ||Fri, May 8 2009, 1:10PM|
The simplest cluster algebra is that associated to triangulations of a
polygon. The cluster variables correspond to diagonals of the polygon, and
the clusters are the collections of cluster variables corresponding to
diagonals that fit together to make a triangulation. We will mainly be
interested in two features of this setup: the tropical cluster algebra (as
in work of Gekhtman-Shapiro-Vainstein and Fomin-Thurston) which is the
tropicalization of the usual cluster algebra, and the connection to the
tilting theory of the path algebra of a linearly oriented path.
I will discuss an analogous situation, in which the polygon is replaced by a
2d-dimensional cyclic polytope. The analogues of the cluster variables are
the d-dimensional internal faces of the polytope. I will describe an
analogue of the tropical cluster algebra on this set of variables. I will
also describe (briefly) the connection to tilting theory, in which the
linearly oriented path algebra is replaced by one of its higher Auslander
algebras (as constructed by Iyama).
This is joint work with Steffen Oppermann (NTNU).