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39
Clustertilted algebras are Gorenstein and stably
 CalabiYau, Adv. Math
"... Abstract. We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that ..."
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Abstract. We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that clustertilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [27]. In addition, we prove a general result about relative 3CalabiYau duality over non stable endomorphism rings. This strengthens and generalizes the Extgroup symmetries obtained in [27] for simple modules. Finally, we generalize the results on relative CalabiYau duality from 2CalabiYau to dCalabiYau categories. We show how to produce many examples of dcluster tilted algebras. 1.
CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
"... Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). I ..."
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Cited by 113 (6 self)
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Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences. Contents
A geometric description of mcluster categories
"... Abstract. We show that the mcluster category of type An−1 is equivalent to ..."
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Cited by 39 (5 self)
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Abstract. We show that the mcluster category of type An−1 is equivalent to
Defining an m−cluster category
, 2005
"... Abstract. We show that a certain orbit category considered by Keller encodes the combinatorics of the mclusters 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. T ..."
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Cited by 35 (2 self)
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Abstract. We show that a certain orbit category considered by Keller encodes the combinatorics of the mclusters 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 AuslanderReiten translate τ.) The cluster category C(H) is a triangulated KrullSchmidt 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
Cyclic sieving and noncrossing partitions for complex reflection groups
, 2007
"... We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for wellgenerated complex reflection groups.
..."
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Cited by 31 (2 self)
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We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for wellgenerated complex reflection groups.
Generalized cluster complexes via quiver representations
 DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NEW
, 2007
"... We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using d−cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d−compa ..."
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We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using d−cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d−compatibility degree (−−) on any pair of “colored ” almost positive real Schur roots which generalizes previous definitions on the noncolored case, and call two such roots compatible provided the d−compatibility degree of them is zero. Associated to the root system Φ corresponding to the valued quiver, by using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and d−compatible subsets as simplicies. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading.
mcluster categories and mreplicated algebras
, 2008
"... Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the mcluster category Cm(A) of A is the mleft part Lm(A (m) ) of the mreplicated algebra of A. Moreover, we obtain a onetoone correspondence between the tilting objects in Cm(A) (that ..."
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Cited by 20 (1 self)
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Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the mcluster category Cm(A) of A is the mleft part Lm(A (m) ) of the mreplicated algebra of A. Moreover, we obtain a onetoone correspondence between the tilting objects in Cm(A) (that is, the mclusters) and those tilting modules in modA (m) for which all non projectiveinjective direct summands lie in Lm(A (m)). Furthermore, we study the module category of A (m) and show that a basic exceptional module with the correct number of nonisomorphic indecomposable summands is actually a tilting module. We also show how to determine the projective dimension of an indecomposable A (m)module from its position in the AuslanderReiten quiver.
The cyclic sieving phenomenon for faces of generalized cluster complexes, arXiv preprint math.CO/0612679
"... Abstract. The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge’s q = −1 phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin ..."
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Abstract. The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge’s q = −1 phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin and Zelevinsky. In this paper, the faces of various dimensions of the generalized cluster complexes in type An, Bn, Dn, and I2(a) are shown to exhibit the cyclic sieving phenomenon under a cyclic group action. For the cluster complexes of exceptional type E6, E7, E8, F4, H3, and H4, a verification for such a phenomenon on their maximal faces is given. 1.
On the enumeration of positive cells in generalized cluster complexes and Catalan . . .
, 2006
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The F triangle of the generalised cluster complex, Topics in discrete mathematics
 Algorithms Combin
, 2006
"... Abstract. The Ftriangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the Ftriangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the “M = F Conjecture ” of Armstrong which predicts a surprisi ..."
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Cited by 12 (3 self)
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Abstract. The Ftriangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the Ftriangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the “M = F Conjecture ” of Armstrong which predicts a surprising relation between the Ftriangle and the Möbius function of his mdivisible partition poset associated to a finite root system. 1. Introduction. Fomin and Zelevinsky created a new exciting research field when they invented cluster algebras in [11]. The classification of cluster algebras of finite type from [12] says that there is a onetoone correspondence between finitetype cluster algebras and finite root systems. Furthermore, for each finite root system Φ, Fomin and Zelevinsky [13] defined a simplicial complex corresponding to the associated cluster algebra,