Noncrossing partitions and the shard intersection order

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. © 2010 Springer Science+Business Media, LLC.