Let Q be an acyclic quiver and s be a sequence with elements in the vertex set Q_0. We describe an induced sequence of simple (backward) tilting in the bounded derived category D(Q), starting from the standard heart H_Q = modkQ and ending at another heart H_s in D(Q). Then we show that s is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to HQ[−1]; it is maximal if and only if H_s = H_Q[−1]. This provides a categorical way to understand green mutations. Further, fix a Coxeter element c in the Coxeter group W_Q of Q, which is admissible with respect to the orientation of Q. We prove that the sequence \tilde{w} induced by a c-sortable word w is a green mutation sequence. As a consequence, we obtain a bijection between c-sortable words and finite torsion classes in H_Q. As byproducts, the interpretations of inversions, descents and cover reflections of a c-sortable word w are given in terms of the combinatorics of green mutations.

We study the cluster combinatorics of d-cluster tilting objects in d-cluster categories. Using mutations of maximal rigid objects in d-cluster categories, which are defined in a similar way to mutations for d-cluster tilting objects, we prove the equivalences between d-cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of d+1 triangles of d-cluster tilting objects in [O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (1) (2008) 117–168], we prove that any almost complete d-cluster tilting object has exactly d+1 complements, compute the extension groups between these complements, and study the middle terms of these d+1 triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established for d-cluster categories in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006) 572–618]. They are applied to the Fomin–Reading generalized cluster complexes of finite root systems defined and studied in [S. Fomin, N. Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not. 44 (2005) 2709–2757; H. Thomas, Defining an m-cluster category, J. Algebra 318 (2007) 37–46; K. Baur, R. Marsh, A geometric description of m-cluster categories, Trans. Amer. Math. Soc. 360 (2008) 5789–5803; K. Baur, R. Marsh, A geometric description of the m-cluster categories of type D_n, preprint, arXiv:math.RT/0610512; see also Int. Math. Res. Not. 2007 (2007), doi:10.1093/imrn/rnm011], and to that of infinite root systems [B. Zhu, Generalized cluster complexes via quiver representations, J. Algebraic Combin. 27 (2008) 25–54].

We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube C_n of rank n. For any indecomposable rigid object M in C_n, we define an analogous X_M of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of M. We show that X_M,X_{M′} satisfy the mutation formula when M,M′ form an exchange pair, and that X_? : M↦X_M gives a bijection from the set of indecomposable rigid objects in C_n to the set of cluster variables of cluster algebra of type C_{n−1}, which induces a bijection between the set of basic maximal rigid objects in C_n and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube Cn encode the combinatorics of the cluster algebra of type B_{n−1} since the combinatorics of cluster algebras of type B_{n−1} or of type C_{n−1} are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.

We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type A_n or A_\infty is given via rotation of Ptolemy diagrams.

In this paper, we give a complete classification of cotorsion pairs in a cluster category C of type A^\infty_\infty via certain configurations of arcs, called τ-compact Ptolemy diagrams, in an infinite strip with marked points. As applications, we classify t-structures and functorially finite rigid subcategories in C, respectively. We also deduce Liu-Paquette's classification of cluster tilting subcategories of C and Ng's classification of torsion pairs in the cluster category of type A^\infty_\infty.