Ordered exchange graphs

Abstract

Abstract. The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in nonpositive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support τ -tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs studied in this article can be considered as a generalization or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of this article is to axiomatize the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other.

Tagged mapping class groups: Auslander–Reiten translation

“…Moreover, we denote by CEG * (S) the exchange graph of reachable cluster tilting sets of C (S), that is, the connected unoriented graph whose vertices are the reachable cluster tilting sets and whose edges are the mutations. We refer to [4] for more details on and equivalent definitions of the graph CEG * (S). Let C × (S) be the set consisting of objects that appear in some cluster tilting set P in CEG * (S).…”

Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions

“…More recent work (see [9] and the many references therein) has shown that exchange graphs of many cluster algebras can be modeled using many different representation theoretic objects related to certain Jacobian algebras Λ [21]. In particular, the poset of functorially finite torsion pairs in the module category of Λ and the poset of 2-term simple-minded collections in the bounded derived category of Λ are isomorphic to the oriented exchange graph [7] of the cluster algebra defined by the quiver of Λ.…”

“…torsion-free part) generated (resp. cogenerated) by indecomposable Λ-modules M with dimpM q (resp.´dimpM q) a row vector of C (see [9] for a proof that these maps are bijections when Λ is a finite dimensional Jacobian algebra).…”

Minimal length maximal green sequences

“…The set c-matpQq can be regarded as a directed graph whose vertices are c-matrices and whose directed edges are exactly those of the form C Q Ñ C µ k Q where c k P C Q is positive. Now by regarding c-matpQq as a directed graph, it follows from [7] that Ý Ý Ñ EGp p Qqc-matpQq and maximal directed paths in each are in natural bijection. 1 Moreover, each directed edge C Q Ñ C µ k Q of c-matpQq is labeled by the c-vector c k .…”

“…Perhaps surprisingly, the c-matrices and g-matrices of a quiver can be obtained from each other by a simple bijection, as the following theorem shows. [7,Theorem 4.17]) The map c-matpQq ÝÑ g-matpQq C Þ ÝÑ pC´1q T pG´1q T ÐÝ G sending a c-matrix or g-matrix to its inverse transpose is a bijection, and this map commutes with mutation of c-matrices and g-matrices of Q.…”