Noncrossing partitions and the shard intersection order

Abstract

We define a new lattice structure 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 the shard intersection order, like Mobius 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 the shard intersection order (i.e. chains in the shard intersection order) 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 shard intersection order 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.

44 pages, 15 figures. Version 2: Small changes to belatedly bring this into closer agreement with the last pre-publication version

References in corpus (16)

• Clusters, Coxeter-sortable elements and noncrossing partitions
• Generalized cluster complexes and Coxeter combinatorics
• Cambrian Lattices
• Noncrossing partitions and representations of quivers
• Cambrian fans
• Sortable elements and Cambrian lattices
• Lattice congruences, fans and Hopf algebras
• Lattice congruences of the weak order
• Sortable elements in infinite Coxeter groups
• Cluster Complexes via Semi-Invariants
• Permutohedra, associahedra, and beyond
• The cd-index of Bruhat intervals
• Chains in the noncrossing partition lattice
• The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
• From Permutahedron to Associahedron
• Permutahedra and generalized associahedra

Cited by in corpus (12)

• The Core Label Order of a Congruence-Uniform Lattice
• Absolute order in general linear groups
• Counting chains in the noncrossing partition lattice via the W-Laplacian
• The Shard Intersection Order on Permutations
• Hochschild lattices and shuffle lattices
• The fundamental theorem of finite semidistributive lattices
• Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group
• The category of a partitioned fan
• Noncrossing arc diagrams and canonical join representations
• Coxeter-biCatalan combinatorics
• Absolute order and involutions
• Bubble Lattices I: Structure