Companion Bases for Cluster-Tilted Algebras

Abstract

Motivated by work of Barot, Geiss and Zelevinsky, we study a collection of Z-bases (which we call companion bases) of the integral root lattice of a root system of simply-laced Dynkin type. Each companion basis is associated with the quiver of a cluster-tilted algebra of the corresponding type.In type A, we establish that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra may be obtained, up to sign, by expanding the positive roots in terms of any companion basis for the quiver of that algebra. This generalises part of Gabriel's Theorem.Also, we describe the relationship between different companion bases for the same quiver and show how to mutate a companion basis for a quiver to produce a companion basis for a mutation of that quiver.

Reflection group presentations arising from cluster algebras

“…In the general case (not necessarily simply-laced), we use A ij = (β i , β ∨ j ). Parsons uses companion bases to obtain the dimension vectors of the indecomposable modules over the cluster-tilted algebra associated to (x, B) by [3] and [4], in the type A case (they are obtained by expanding arbitrary roots in terms of the basis; see [10,Theorem 5.3]). We remark that independent later work of Ringel [11] also obtains such dimension vectors (in a more general setting which includes all the finite type cases).…”

“…. , β n } is a companion basis for an exchange matrix B in a cluster algebra of finite type as above, and fix 1 ≤ k ≤ n. Parsons [10,Thm. 6.1] defines the (inward) mutation B ′ = µ k (B) of B at k to be the subset {β ′ 1 , β ′ 2 , .…”

“…This work was mainly motivated by the notions of quasi-Cartan companion [1] and companion basis [10,Defn. 4.1] (see also [9]).…”

“…This means that the diagonal entries are equal to 2, that the values of the off-diagonal entries in A coincide with the corresponding entries of B up to sign, and that the symmetrisation of the matrix is positive definite. Quasi-Cartan companions are used in [1] to give criteria for whether a cluster algebra is of finite type and companion bases were used in [9,10] to describe the dimension vectors of indecomposable modules over cluster-tilted algebras in type A (dimension vectors of indecomposable modules over cluster-tilted algebras in the general concealed case were also independently calculated later in [11]). …”

Explicit Construction of Companion Bases

“…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.…”