OIST Representation Theory Seminar

Nicolas Williams, Lancaster University

Title: Higher-dimensional cluster combinatorics and representation theory

Abstract: Perhaps the most prominent example of a cluster algebra is the type A cluster algebra, where clusters are in bijection with triangulations of a convex polygon, as observed by Fomin and Zelevinsky. A categorical version of this relationship is that triangulations of a convex polygon are in bijection with cluster-tilting objects in the cluster category of the path algebra of the type A quiver. In each case, mutating the cluster or cluster-tilting object corresponds to flipping a diagonal inside a quadrilateral. It is natural to wonder whether any similar relationship exists for triangulations of higher-dimensional polytopes. Indeed, in a beautiful paper Oppermann and Thomas show that triangulations of even-dimensional cyclic polytopes are in bijection with cluster-tilting objects in the cluster categories of the higher Auslander algebras of type A, which were introduced by Iyama. Mutating the cluster-tilting objects corresponds to bistellar flips of triangulations, which are the higher-dimensional analogues of flipping a diagonal inside a quadrilateral. In this talk, we will outline the work of Oppermann and Thomas, and explain the odd-dimensional half of the picture too. Indeed, the speaker has shown that triangulations of odd-dimensional cyclic polytopes are in bijection with equivalence classes of maximal green sequences for the higher Auslander algebras of type A, where maximal green sequences are maximal chains of cluster-tilting objects.