Higher-dimensional cluster combinatorics and representation theory

Abstract

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 occuring in the mutation of cluster tilting objects.

Cluster tilting objects in generalized higher cluster categories

“…Let ⟨A⟩ B be the thick subcategory of D b (B) generated by the image of p * . We call the triangulated hullC A = ⟨A⟩ B /perB of O A the (n − 1)-cluster category of A.Here we would like to point out that the authors of[21] make use of the d-Calabi-Yau generalized cluster categories, whereas they do not give any explicit proof for their construction. In general, this category C A has infinite-dimensional morphism spaces.…”

A Survey of the Higher Stasheff-Tamari Orders

“…Furthermore, this can be done in such a way that basic tilting modules correspond to triangulations, and mutations correspond to diagonal flips. For more on this, see the references in Oppermann and Thomas [26], as well as in Thomas's survey [45] in this volume.…”

“…Oppermann and Thomas [26] recently uncovered a fascinating connection between the representation theory of certain algebras and triangulations of the evendimensional cyclic polytopes C(m, d) with d = 2e. We will not do justice to their results here and refer the interested reader to their paper for more details and precise statements.…”

“…C(n + 2e + 1, 2e)) with certain kinds of indecomposable objects in two different categories constructed from A e n -modules. Furthermore, they do this in such a way that, in each case, basic tilting modules correspond to triangulations, and the appropriate analogues of mutation correspond to bistellar flips [26,Theorems 4.4,6.4].…”

“…Nevertheless, they share many beautiful properties with the Tamari order. Here we survey the work on them by Edelman and Reiner [15], Rambau [31], Reiner [36, §6] Edelman, Rambau and Reiner [14], Thomas [43,44], and most recently, Oppermann and Thomas [26]. We also discuss Jörg Rambau University of Bayreuth, Germany, e-mail: joerg.rambau@uni-bayreuth.de Victor Reiner University of Minnesota, Minneapolis, USA, e-mail: reiner@math.umn.edu work on the closely related Baues problem for subdivisions of cyclic polytopes and zonotopes, as studied by Rambau and Santos [33], Athanasiadis, Rambau and Santos [3], and Athanasiadis [2].…”