Noncrossing partitions and the shard intersection order
by
Nathan Reading
North Carolina State University

I will discuss the shard intersection order on a finite Coxeter group W. This poset is a lattice and has the noncrossing partition lattice NC(W) as a sublattice. This new construction of NC(W) yields a new proof that NC(W) is a lattice. The shard intersection order is graded and atomic. Its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of the shard intersection order, like Möbius number, number of maximal chains, etc., are analogous to corresponding properties of NC(W). The shard intersection order is most naturally defined in terms of the polyhedral geometry of the reflecting hyperplanes of W, and in particular certain codimension-1 polyhedral cones called shards. The reflecting hyperplanes are cut into shards according to a simple rule. Shards were originally defined as a way of understanding lattice congruences of the weak order on W. The collection of arbitrary intersections of shards forms a lattice under reverse containment. Arbitrary intersections of shards are in bijection with elements of W, so the lattice of shard intersections defines a partial order on W, which we call the shard intersection order. I will illustrate the definitions and results with a running example, taking W to be the symmetric group S₄.

Date received: February 20, 2009