The University of Michigan Combinatorics Seminar
We introduce a new partial order (in fact, lattice) Ψ(W) on a finite Coxeter group W, weaker than the weak order and having the noncrossing partition lattice NC(W) as a sublattice. This provides, in particular, a new proof that NC(W) is a lattice. The lattice Ψ(W) is graded and atomic and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of Ψ(W), like Möbius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC(W). Furthermore, viewing NC(W) as a sublattice of Ψ(W) leads to new proofs of the known properties of NC(W).
Shards are certain codimension-1 polyhedral cones that govern the lattice theory of the weak order on W. The reflecting hyperplanes are cut into shards according to a simple rule. The collection of arbitrary intersections of shards forms a lattice under reverse containment. Surprisingly there is a bijection between intersections of shards and elements of W. Furthermore, this bijection is an isomorphism between the lattice of intersections of shards and Ψ(W). The realization in terms of shards is the key to most of what can be proved about Ψ(W). This geometric approach also brings to light close connections between NC(W) and semi-invariants of quivers.
For those less familiar with Coxeter groups, I will illustrate the definitions and results with a running example, taking W to be the symmetric group S4.