MIT Combinatorics Seminar
Polyhedral geometry of Cambrian lattices
David Speyer (University of Michigan)
Friday, December 8, 2006 4:15 pm Room 2136
ABSTRACT

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.


