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32
Cambrian lattices
"... Abstract. For an arbitrary finite Coxeter group W, we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatoriall ..."
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Abstract. For an arbitrary finite Coxeter group W, we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B, we obtain, by means of a fiberpolytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky’s construction of the normal fan as a “cluster fan. ” Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two “Tamari ” lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.
Cambrian Fans
, 2008
"... For a finite Coxeter group W and a Coxeter element c of W, the cCambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the csortable elements of W. The main result of this paper is that the known bijection clc between csorta ..."
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Cited by 25 (7 self)
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For a finite Coxeter group W and a Coxeter element c of W, the cCambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the csortable elements of W. The main result of this paper is that the known bijection clc between csortable elements and cclusters induces a combinatorial isomorphism of fans. In particular, the cCambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the cCambrian fan are generated by certain vectors in the Worbit of the fundamental weights, while the rays of the ccluster fan are generated by certain roots. For particular (“bipartite”) choices of c, we show that the cCambrian fan is linearly isomorphic to the ccluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map clc, on cclusters by the cCambrian lattice. We give a simple bijection from cclusters to cnoncrossing 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
PERMUTAHEDRA AND GENERALIZED ASSOCIAHEDRA
"... Abstract. Given a finite Coxeter system (W, S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W, we construct a simple polytope whose outer normal fan is N. Reading’s Cambrian fan Fc, settling a conjecture of Reading that this is possible. We call this polytope the c ..."
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Cited by 23 (6 self)
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Abstract. Given a finite Coxeter system (W, S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W, we construct a simple polytope whose outer normal fan is N. Reading’s Cambrian fan Fc, settling a conjecture of Reading that this is possible. We call this polytope the cgeneralized associahedron. Our approach generalizes Loday’s realization of the associahedron (a type A cgeneralized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the csingleton cones, the cones in the cCambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if W is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to W, then we show that our realizations have
Lattice Congruences of the Weak Order
 ORDER
, 2004
"... We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of joinirreducibles of the congruence la ..."
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Cited by 18 (8 self)
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We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of joinirreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K ⊆ S, letηK: w ↦ → wK be the projection onto the parabolic subgroup WK. We show that the fibers of ηK constitute the smallest lattice congruence with 1 ≡ s for every s ∈ (S − K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of joinirreducibles of the congruence lattice of the weak order.
Sortable elements in infinite Coxeter groups
 Trans. Amer. Math. Soc
"... Abstract. In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In ..."
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Abstract. In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finitetype proofs into uniform proofs (rather than type by type proofs), generalize many of the finitetype results and prove new and more refined results. The key tools in our proofs include a skewsymmetric form related to (a generalization of) the Euler form of quiver theory and the projection πc ↓ mapping each element of W to the unique maximal csortable element below it in the weak order. The fibers of πc ↓ essentially define the cCambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of πc ↓. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans. Contents
Combinatorial frameworks for cluster algebras
 Int. Math. Res. Not., published online May
, 2015
"... Abstract. We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model incorporating information about exchange matrices, principal c ..."
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Abstract. We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model incorporating information about exchange matrices, principal coefficients, gvectors, gvector fans, and (conjecturally) denominator vectors. The idea behind frameworks arises from Cambrian combinatorics and sortable elements, and in this paper, we use sortable elements to construct a framework for any cluster algebra with an acyclic initial exchange matrix. This Cambrian framework yields a model of the entire (principal coefficients) exchange graph when the cluster algebra is of finite type. Outside of finite type, the Cambrian framework models only part of the exchange graph. In a forthcoming paper, we extend the Cambrian construction to produce a complete framework for a
Polyhedral models for generalized associahedra via coxeter elements
, 2011
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Algebraic and combinatorial structures on Baxter permutations
 In FPSAC 2011, DMTCS proc. AO
, 2011
"... Abstract. We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasisymmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees, etc.). This construction relies on the definition of ..."
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Cited by 7 (1 self)
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Abstract. We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasisymmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees, etc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a RobinsonSchenstedlike insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras, Journal of Combinatorial Theory Series A, 110:237–273, 2005]. Résumé. Nous proposons une nouvelle construction d’une sousalgèbre de Hopf de l’algèbre de Hopf des fonctions quasisymétriques libres dont les bases sont indexées par les objets de la famille combinatoire de Baxter (i.e. permutations de Baxter, couples d’arbres binaires jumeaux, etc.). Cette construction repose sur la définition du monoïde de Baxter, analogue du monoïde plaxique et du monoïde sylvestre, et d’un algorithme d’insertion analogue à l’algorithme de RobinsonSchensted. Les propriétés algébriques de cette algèbre de Hopf sont étudiées. Cette algèbre de Hopf est apparue pour la première fois dans le travail de Reading [Lattice congruences, fans and Hopf algebras, Journal of
NONCROSSING PARTITIONS AND THE SHARD INTERSECTION ORDER
, 2009
"... We define a new lattice structure (W, ≼) on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new construction of NC(W) yields a new proof that NC(W) is a la ..."
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We define a new lattice structure (W, ≼) on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new construction of NC(W) yields a new proof that NC(W) is a lattice. The shard intersection order is graded and its rank generating function is the WEulerian polynomial. Many ordertheoretic properties of (W, ≼), like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC(W). There is a natural dimensionpreserving bijection between simplices in the order complex of (W, ≼) (i.e. chains in (W, ≼)) and simplices in a certain pulling triangulation of the Wpermutohedron. Restricting the bijection to the order complex of NC(W) yields a bijection to simplices in a pulling triangulation of the Wassociahedron. The lattice (W, ≼) is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.