MIT Combinatorics Seminar

Polyhedral geometry of Cambrian lattices

David Speyer (University of Michigan)

Friday, December 8, 2006   4:15 pm    Room 2-136


Let $W$ be a finite Coxeter group of rank $n$ and $c$ a coxeter element. The $c$-clusters of $W$ are certain $n$ element collections of roots of $W$, whose definition is motivated by the theory of cluster algebras. The $c$-noncrossing partitions are certain elements of $W$ which form a rank symmetric lattice. The $c$-Cambrian lattice is a certain lattice quotient of the weak order on $W$, defined by Nathan Reading. Nathan Reading has constructed bijections between the $c$-clusters, the $c$-noncrossing partitions and the elements of the $c$-Cambrian lattice. In this talk, based on joint work with Nathan Reading, I will describe a polyhedral picture that makes all of these bijections transparent to describe. Specifically, elements of the $c$- cambrian lattice can be thought of as maximal cones in a certain coarsening of the $W$-hyperplane arrangement. We show that these cones are simplicial and that the simplicial complex they form is combinatorially, though not metrically, equivalent to the cluster complex. The invariant subspaces of noncrossing partitions are the bottom faces of cones in the Cambrian lattice. This picture allows us to assign geometric meanings to several other combinatorial objects from the theory of cluster algebras, such as $g$-vectors and quasi- Cartan companion matrices. At the end, I will sketch some hopes about how the theory might extend to infinite Coxeter groups.