Abstract

Speaker: Nathan Reading, (University of Michigan)

Title: Noncrossing partitions and the Coxeter plane

Abstract:

Classical noncrossing partitions are partitions p such that a certain planar diagram of p has non-intersecting parts. These form a lattice under refinement order. Algebraically, noncrossing partitions relate to factorizations of an n-cycle in S_ⁿ into transpositions. The construction can be generalized to an arbitrary finite Coxeter group W (i.e. a finite group generated by reflections) by considering factorizations of a Coxeter element c into reflections. One obtains a "noncrossing partition lattice" for W which is instrumental in constructing finite K(pi,1) spaces and monoid structures for the Artin group associated to W.

In this talk we study noncrossing partitions by way of the Coxeter plane, a certain plane fixed (as a set) by the action of c. We give a general recursive formula for counting maximal chains in noncrossing partition lattices. We also show how the planar diagrams for classical noncrossing partitions and their counterparts of types B and D arise from a uniform construction, rather than in an ad hoc manner and discuss this construction in the case of the exceptional groups.

The talk requires no prior knowledge, on the part of the audience, of Coxeter groups or noncrossing partitions.

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