Cambrian Lattices

By Nathan Reading. 2004

248 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f463467e888f6301e564c30e295a97c1de899ebc

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Citations in the corpus

Non-kissing complexes and tau-tilting for gentle algebras (2017, background): “plexes and accordion complexes. (iii) Path quivers: When the quiver is a path, the non-kissing complex is a simplicial associahedron, the non-kissing lattice is a type ACambrian lattice of N. Reading [Rea06] (including the classical Tamari lattice), and the non-kissing associahedron is an associahed”
Sortable elements in infinite Coxeter groups (2008, background): “ations 38 9. The Cambrian fan 42 10. Rank-three examples 47 Acknowledgments 57 References 57 1 2 NATHAN READING AND DAVID E SPEYER 1. Introduction This paper is the continuation of a series of papers [31, 32, 33, 34] devoted to the study of sortable elements and Cambrian lattices. The primary motiva”
Sur le nombre d'intervalles dans les treillis de Tamari (2006, background): “[8] Nathan Reading.”
Powers of Coxeter elements in infinite groups are reduced (2007, background): “in [5]; it is the opposite of the one used in [10] and [11].”
Parking Spaces (2012, background): “These can be defined by (3.2) NarW (t) := ∑ X∈NC(W ) t dimC(X) = ∑ w∈[1,c]T t dimV w KirkW (t) := NarW (t+ 1) := ∑ C t n−|C| where in the last sum, C ranges over all clusters in the cluster complex of finite type associated to W by Fomin and Zelevinsky [23] in the crystallographic case, or over all ”
Noncrossing partitions and the shard intersection order (2009, unknown): “This fact has never been published, but has been observed independently by several researchers in the brief time since the results of [32] were disseminated.”
Universal geometric cluster algebras (2012, unknown): “The definitions and results given here are inspired by the juxtaposition of several results on cluster algebras of finite type from [12,19,23,30].”
Subword complexes, cluster complexes, and generalized multi-associahedra (2011, background): “Reading constructed c-Cambrian fans, which are complete simplicial fans coarsening the Coxeter fan, see [47].”
Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups (2006, background): “In a development parallel to the theory of cluster algebras, Reading has defined a class of lattice quotients of the weak order on W , which he calls Cambrian lattices [102].”
Permutahedra and generalized associahedra (2007, methodology): “Reading shows that the c-sortable elements constitute a sublattice of the weak order of W which is called the c-Cambrian lattice, [18, 20].”
Brick polytopes of spherical subword complexes and generalized associahedra (2011, background): “ous section, we are now in the position to describe the Cambrian lattice and fan in this new context. The c-sortable elements of W form a sublattice and a quotient lattice of the weak order on W, see [Rea06]. This lattice is called c-Cambrian lattice. The downward projection used to construct the qu”
Coxeter submodular functions and deformations of Coxeter permutahedra (2019, background): “Reading [37] introduced the Cambrian fan Fc, a complete fan with rich combinatorial properties and close connections with the theory of cluster algebras [38].”
Lattice structure of torsion classes for path algebras (2013, background): “N. Reading, ‘Cambrian lattices’, Adv.”
Lattice Properties of Oriented Exchange Graphs and Torsion Classes (2015, background): “For example, if Q is an orientation of a Dynkin diagram of type A, D, or E, then its oriented exchange graph is a Cambrian lattice of the same type (see [26]).”