Cambrian lattices.
type: Item
instance of: scholarly article
title: Cambrian lattices.
zbMATH Open document ID: 1106.20033
author: Nathan Reading
published in: Advances in Mathematics
publication date: 2006-09-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0402086
review text: Let \((W,S)\) be a Coxeter system. The elements of \(S\) are called simple reflections and conjugates of simple reflections are called reflections. The left inversion set of \(w\), denoted \(I(w)\), is the set of all (not necessarily simple) reflections \(t\) such that \(l(tw)<l(w)\). The length \(l(w)\) is equal to \(|I(w)|\). The right weak order on \(W\) is the partial order which sets \(v\leq w\) if and only if \(I(v)\subset I(w)\). The term ``weak order'' always refers to the right weak order. For an arbitrary finite Coxeter group \(W\), the author defines the family of Cambrian lattices for \(W\) as quotients of the weak order on \(W\) with respect to certain lattice congruences. The author associates to each Cambrian lattice a complete fan. In types \(A\) and \(B\) the author obtains, 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, the author proves 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''. The author also shows that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 05E15
zbMATH DE Number: 5056850
zbMATH Keywords: clusters
zbMATH Keywords: fiber polytopes
zbMATH Keywords: generalized associahedra
zbMATH Keywords: lattice congruences
zbMATH Keywords: lattice quotients
zbMATH Keywords: Tamari lattices
zbMATH Keywords: Cambrian lattices
zbMATH Keywords: posets of regions
zbMATH Keywords: simplicial hyperplane arrangements
zbMATH Keywords: weak orders
MaRDI profile type: Publication
OpenAlex ID: W4212973280
arXiv ID: math/0402086
cites work: Fiber polytopes
cites work: Iterated fiber polytopes
cites work: Q3724546
cites work: Shellable nonpure complexes and posets. II
cites work: On the self-linking of knots
cites work: Cayley lattices of finite Coxeter groups are bounded
cites work: Congruences in ordered sets
cites work: Polytopal Realizations of Generalized Associahedra
cites work: Congruence normality: The characterization of the doubling class of convex sets
cites work: Cluster algebras. II: Finite type classification
cites work: \(Y\)-systems and generalized associahedra
cites work: Q4834373
cites work: On the distributivity of a lattice of lattice-congruences
cites work: On Tamari lattices
cites work: Q4165427
cites work: Q4003858
cites work: Q4303668
cites work: Realization of the Stasheff polytope
cites work: Generalized associahedra via quiver representations
cites work: Order dimension, strong Bruhat order and lattice properties for posets
cites work: Lattice congruences of the weak order.
cites work: Lattice congruences, fans and Hopf algebras.
cites work: Clusters, Coxeter-sortable elements and noncrossing partitions
cites work: Sortable elements and Cambrian lattices.
cites work: Non-crossing partitions for classical reflection groups
cites work: Equivariant fiber polytopes
cites work: Combinatorial statistics on type-B analogues of noncrossing partitions and restricted permutations
cites work: A type-B associahedron.
cites work: Q5688183
cites work: The patterns of permutations
DOI: 10.1016/J.AIM.2005.07.010
description: scientific article