Higher-dimensional cluster combinatorics and representation theory

Abstract

Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite-dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects. We study triangulations of even-dimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type $A$ which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any $d$-representation finite algebra we introduce a certain $d$-dimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type $A$ we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogues of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occurring in the mutation of cluster tilting objects.