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 fiber-polytope 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.

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 join-irreducibles 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 join-irreducibles of the congruence lattice of the weak order.

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. 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-generalized associahedron. Our approach generalizes Loday’s realization of the associahedron (a type A c-generalized 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 c-singleton cones, the cones in the c-Cambrian 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

This proposal is about combinatorial algebra, with a geometrical flavor. Together with students and collaborators I have been developing a diverse family of graded algebras and coalgebras, modules and comodules, often with Hopf and differential structures. Their common feature is that all are based upon recursive sequences of convex polytopes. The key point of interest in each case is to see how the algebraic structure reflects the combinatorial structure, and vice versa. We are building upon the foundations laid by many other researchers, especially Gian-Carlo Rota, who most clearly saw the strength of this approach. The historical examples of Hopf algebras SSym and QSym, the Malvenuto-Reutenauer Hopf algebra and the quasisymmetric functions, can be defined using graded bases of permutations and boolean subsets respectively. Loday and Ronco used the fact that certain binary trees can represent both sorts of combinatorial objects to discover the Hopf algebra YSym lying between them. Chapoton capitalized on the fact that the three graded bases could actually be described as the vertex sets of polytope sequences, and defined larger algebras on the faces of the permutohedra, associahedra and cubes. The polytope sequences we study include those familiar examples as well as newer families such as the graph multiplihedra and composihedra. Simultaneously with our study of Hopf algebras we

In 2008 several researchers made exciting advances in the application of algebra and combinatorics to particle physics. The crucial mathematical ingredients included Hopf Algebras and subalgebras, in regard to the specific physical principle of renormalization in quantum electrodynamics and chromodynamics. Renormalization refers to the addition of counterterms to a sequence of divergent integrals for probability

Abstract. We introduce Coxeter-sortable elements of a Coxeter group W. For finite W, we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in terms of their inversion sets and, in the classical

Abstract. 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 W-Eulerian polynomial. Many order-theoretic 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 dimension-preserving bijection between simplices in the order complex of (W, ≼) (i.e. chains in (W, ≼)) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC(W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. 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.