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51
Lattice congruences, fans and Hopf algebras
 J. Combin. Theory Ser. A
"... Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak or ..."
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Cited by 17 (8 self)
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Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the MalvenutoReutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of patternavoidance. Applying these results, we build the MalvenutoReutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of noncommutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations. 1.
ON LINEAR TRANSFORMATIONS PRESERVING THE PÓLYA FREQUENCY PROPERTY
"... We prove that certain linear operators preserve the Pólya frequency property and realrootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and ReinerWelker. ..."
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Cited by 14 (4 self)
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We prove that certain linear operators preserve the Pólya frequency property and realrootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and ReinerWelker.
Cyclic sieving and noncrossing partitions for complex reflection groups, preprint
"... Abstract. We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for wellgenerated complex reflection groups. 1. ..."
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Cited by 14 (1 self)
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Abstract. We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for wellgenerated complex reflection groups. 1.
Cluster algebras IV: coefficients
"... 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 ” coe ..."
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Cited by 14 (1 self)
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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 multigradings 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 Ysystems. We establish a Laurent phenomenon for such Ysystems, 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
Nested complexes and their polyhedral realizations
 Pure and Applied Mathematics Quarterly
"... This note which can be viewed as a complement to [9], presents a selfcontained 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 strikin ..."
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Cited by 14 (0 self)
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This note which can be viewed as a complement to [9], presents a selfcontained 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
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 13 (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
hvectors of generalized associahedra and noncrossing partitions,” arXiv/math.CO/0602293
"... Abstract. A casefree proof is given that the entries of the hvector 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 hvector of ..."
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Cited by 12 (5 self)
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Abstract. A casefree proof is given that the entries of the hvector 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 hvector 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.
Polygon dissections and some generalizations of cluster complexes
, 2005
"... 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 face ..."
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Cited by 11 (3 self)
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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 hvector 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 CohenMacaulay. 1. Introduction and
On the enumeration of positive cells in generalized cluster complexes and Catalan . . .
, 2006
"... ..."
Sortable elements and Cambrian lattices
"... Abstract. We show that the Coxetersortable elements in a finite Coxeter group W are the minimal congruenceclass 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 Coxetersort ..."
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Cited by 10 (5 self)
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Abstract. We show that the Coxetersortable elements in a finite Coxeter group W are the minimal congruenceclass 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 WCatalan combinatorics arising in the context of the lattice theory of the weak order on W. Contents