Explicit Construction of Companion Bases

Abstract

A companion basis for a quiver mutation equivalent to a simplylaced Dynkin quiver is a subset of the associated root system which is a ‫-ޚ‬basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of . It is known in type A (and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation type A or D.2010 Mathematics Subject Classification. 13F60, 05E10, 17B22. Introduction.The cluster algebras of finite type were shown to have a classification by Dynkin diagrams in [12]. We consider the cluster algebra A associated to a simply-laced Dynkin diagram . It follows from the classification result [12, Theorem 1.8] that the exchange matrices of A are skew-symmetric integer matrices with entries not exceeding 1 in absolute value (note that skew-symmetry is preserved under matrix mutation). These exchange matrices can therefore be represented as quivers, which we shall refer to as the quivers of mutation type . Fix such a quiver and denote its set of vertices by 0 . In [16] (originally in [15]), motivated by work in [1], we introduced the notion of a companion basis for as a certain subset of the corresponding root system of type . Specifically, a companion basis for is a subset {γ x : x ∈ 0 } ⊆ whose elements form a ‫-ޚ‬basis for the integral root lattice ‫ޚ‬ and such that for distinct vertices x, y ∈ 0 , (γ x , γ y ) is equal to the number of edges joining x and y in the underlying unoriented graph for .This article continues the study of companion bases initiated in [16]. The focus here is on producing explicit companion bases for quivers. In [16], we already presented a method for finding a companion basis for any given quiver of simply-laced Dynkin mutation type. Indeed, we noted there that a simple system of a root system of simplylaced Dynkin type is a companion basis for any orientation of the associated Dynkin diagram. Furthermore, we introduced a companion basis mutation procedure that, given a companion basis for a quiver of simply-laced Dynkin mutation type, produces a companion basis for any mutation of that quiver. While this, at least in theory, enables us to find a companion basis for any quiver of simply-laced Dynkin mutation type, it has a drawback that can make it difficult to apply in practice. Namely, that it requires us to find a sequence of quiver mutations taking us from a quiver for which we already have a companion basis to the quiver for which we desire to produce one. Here, we