The University of Michigan Combinatorics Seminar
Fall 2003
September 12, 4:10-5:00, 3866 East Hall





Cambrian Lattices

Nathan Reading

University of Michigan




Abstract

Given a finite Coxeter group W, we define a family of "Cambrian lattices" associated to W. Each of these lattices is defined as the quotient of the weak order on W, mod a certain natural congruence. Equivalently, each is the image of the weak order under a lattice homomorphism. When W is the symmetric group, the Tamari lattice is one of the Cambrian lattices associated to W, and when W is B_n, there is a Cambrian lattice which might be called the type B Tamari lattice. In types A and B, the Cambrian lattices arise from a(n) (equivariant) fiber-polytope construction, leading to a fairly complete combinatorial description of these lattices. Work on the combinatorics in type D is in progress. For W of general type, one particular Cambrian lattice might be called the "cluster lattice" due to a close connection to the construction of clusters by Fomin and Zelevinsky.

In this talk, I will define Cambrian lattices and develop the combinatorial description for type A. Time permitting, I will say something about type B, and/or the connection to clusters. I will not assume (nor will I impart) any knowledge of fiber polytopes. I will assume a very basic knowledge of Coxeter groups and the weak order, and in particular the weak order on permutations and signed permutations. I will also assume the definition of a lattice and of a join-irreducible element of a lattice. Those not familiar with these concepts may want to attend the Pre-Cambrian talk where I will provide a very quick overview.


VIGRE Pre-talk

The above talk will be preceded by a VIGRE pre-talk, also by Nathan Reading, to be held from 3:15-3:45pm in 3866 East Hall, i.e. the regular seminar room. The topic of the pre-talk is:

A quick, very basic overview of Coxeter groups, weak order and lattices.