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Cyclic polytopes, analogues of tropical cluster algebras, and higher Auslander algebras

Algebra & 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).