UK WILDCATS Seminar

"Noncrossing partitions and intersections of shards"

Nathan Reading
North Carolina State University

Monday, November 3, 2008
4:00 pm, 238 Classroom Building

Abstract:

I will discuss a new partial order (in fact, lattice) Psi(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 Psi(W) is graded and atomic and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of Psi(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 Psi(W) leads to new proofs of the known properties of NC(W). The lattice Psi(W) is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of 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. This bijection induces a lattice structure Psi(W) on W. 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 S_4.

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