Abstract. We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of “principal ” coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parametrizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parametrizations, some proven and some conjectural, suggest links to duality conjectures of V. Fock and A. Goncharov. The coefficient dynamics leads to a natural generalization of Al. Zamolodchikov’s Y-systems. We establish a Laurent phenomenon for such Y-systems, previously known in finite type only, and sharpen the periodicity result from an earlier paper. For cluster algebras of finite type, we identify a canonical “universal ” choice

Abstract. Let W be a Weyl group corresponding to the root system An−1 or Bn. We define a simplicial complex ∆m W in terms of polygon dissections for such a group and any positive integer m. For m = 1, ∆m W is isomorphic to the cluster complex corresponding to W, defined in [8]. We enumerate the faces of ∆m W and show that the entries of its h-vector are given by the generalized Narayana numbers Nm W (i), defined in [3]. We also prove that for any m≥1 the complex ∆m W is shellable and hence Cohen-Macaulay. 1. Introduction and

Abstract. A case-free proof is given that the entries of the h-vector of the cluster complex ∆(Φ), associated by S. Fomin and A. Zelevinsky to a finite root system Φ, count elements of the lattice L of noncrossing partitions of corresponding type by rank. Similar interpretations for the h-vector of the positive part of ∆(Φ) are provided. The proof utilizes the appearance of the complex ∆(Φ) in the context of the lattice L, in recent work of two of the authors, as well as an explicit shelling of ∆(Φ). 1.

Abstract. We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for well-generated complex reflection groups. 1.

Abstract. We show that a certain orbit category considered by Keller encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. This allows us to give typeuniform proofs of certain results of Fomin and Reading in the simply laced cases. For Φ any root system, Fomin and Zelevinsky [FZ] define a cluster complex ∆(Φ), a simplicial complex on Φ≥−1, the almost positive roots of Φ. Its facets (maximal faces) are called clusters. In [BM+], starting in the more general context of a finite dimensional hereditary algebra H over a field K, Buan et al. define a cluster category C(H) = D b (H)/τ −1 [1]. (D b (H) is the bounded derived category of representations of H; more will be said below about it, its shift functor [1], and its Auslander-Reiten translate τ.) The cluster category C(H) is a triangulated Krull-Schmidt category. We will be mainly interested in the case where H is a path algebra associated to the simply laced root system Φ, in which case we write C(Φ) for C(H). There is a bijection V taking Φ≥−1 to the indecomposables of C(Φ). A (cluster)-tilting set

In this paper we investigate a variant of the octahedron recurrence of Robbins-Rumsey [8] called the bounded octahedron recurrence. It was first described by Kamnitzer and the author in [4], where it was used to relate the commutativity isomorphism for gl(n)-crystals with the Schützenberger involution on Young tableaux.

Abstract. We show that the Coxeter-sortable elements in a finite Coxeter group W are the minimal congruence-class representatives of a lattice congruence of the weak order on W. We identify this congruence as the Cambrian congruence on W, so that the Cambrian lattice is the weak order on Coxetersortable elements. These results exhibit W-Catalan combinatorics arising in the context of the lattice theory of the weak order on W. Contents

Abstract. For a finite Coxeter group W and a Coxeter element c of W, the c-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the c-sortable elements of W. The main result of this paper is that the known bijection clc between c-sortable elements and c-clusters induces a combinatorial isomorphism of fans. In particular, the c-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the c-Cambrian fan are generated by certain vectors in the W-orbit of the fundamental weights, while the rays of the c-cluster fan are generated by certain roots. For particular (“bipartite”) choices of c, we show that the c-Cambrian fan is linearly isomorphic to the c-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map clc, on c-clusters by the c-Cambrian lattice. We give a simple bijection from c-clusters to c-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric

Abstract. We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the Bott-Taubes polytope) and compare them to the permutahedron of type A and B respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type An or Bn respectively as only input and which specialises to a procedure presented by J.-L. Loday for a certain orientation of An. 1.

This note which can be viewed as a complement to [9], presents a self-contained overview of basic properties of nested complexes and their two dual polyhedral realizations: as complete simplicial fans, and as simple polytopes. Most of the results are not new; our aim is to bring into focus a striking similarity between nested complexes