Discrete CATS Seminar
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U N I V E R S I T Y O F K E N T U C K Y

DISCRETE
CATS
SEMINAR

WHERE CATS =
COMBINATORICS,
ALGEBRA,
TOPOLOGY
&
STATISTICS!

845 PATTERSON OFFICE TOWER

2008 - 2009

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UK
WILDCATS
Seminar

"Noncrossing partitions and intersections of shards"

Nathan Reading

North Carolina State University

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Monday, November 3, 2008

4:00 pm, 238 Classroom Building

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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.