We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted a...

Broline, Crowe and Isaacs have computed the determinant of a matrix associated to a Conway-Coxeter frieze pattern. We generalise their result to the corresponding frieze pattern of cluster variables arising from the Fomin-Zelevinsky cluster algebra of type A. We give a representation-theoretic interpretation of this result in terms of certain configurations of indecomposable objects in the root...

We extend a T -path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D.

We review some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G. We focus on the case G = SL(N,C) and with M a knot complement. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the ChernSimons path integral. We also review various applications and open...

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH(B) with coefficients in the B-B-bimodule B. We find several consequences when B is representation-finite, and also in the case where B is cluster-tilted of type Ã. 2000 Mathematics Subject Classification : 16E40

We construct frieze patterns of type DN with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type DN , we show that the numbers in the pattern can be interpreted as specialisations of cluster variables in the corresponding Fomin-Zelevinsky cluster ...

Kang, Kashiwara, Kim and Oh have proved that cluster monomials lie in the dual canonical basis, under a symmetric type assumiption. This involves constructing a monoidal categorification of a quantum cluster algebra using representations of KLR algebras. We use a folding technique to generalise their results to all Lie types.

The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps “reachable” indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalises the idea that the cluster category is a “categorification” of the cluster algebra. The definition of the Caldero-C...

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We show that the set of cluster monomials for the cluster algebra of type D4 contains a basis of the Z-module Z[x1,1, . . . , x3,3]. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases....