The University of Michigan Combinatorics Seminar
Fall 2010
September 24, 4:10-5:00, 3866 East Hall

Cyclic polytopes and higher dimensional analogues of tropical cluster algebras

Hugh Thomas

University of New Brunswick


The simplest cluster algebras are those associated to triangulations of a polygon. I will be mainly interested, not in the usual cluster algebras, but in their tropical version, as in work of Gekhtman-Shapiro-Vainshtein and Fomin-Thurston. I will discuss a higher-dimensional generalization of this framework, in which the polygon has been replaced by an even-dimensional cyclic polytope. In this setting, we find an analogue of the tropical cluster algebra. I will discuss various aspects of the usual theory which generalize, and some which don't. I will touch only briefly on the higher-dimensional analogue of the link from cluster algebras to tilting theory of hereditary algebras and the cluster category, since that will be the topic of my talk on Tuesday, September 21, in the Algebra seminar. (The two talks will be independent, and I will try to keep overlap to a minimum.)
This is joint work with Steffen Oppermann, and is mostly based on arXiv:1001.5437.