The University of Michigan Combinatorics Seminar
Fall 2006
September 29, 4:10-5:00, 3866 East Hall

Sortable elements and Cambrian fans

Nathan Reading

North Carolina State University


Let W be a finite Coxeter group. A Cambrian fan associated to W is a certain simplicial fan obtained as a coarsening of the fan defined by the reflecting hyperplanes of W. This fan is interesting because it is combinatorially isomorphic to the fan defined by clusters of almost positive roots. (The latter is the normal fan of the generalized associahedron and its underlying simplicial complex is the combinatorial backbone of a cluster algebra of finite type.) Moreover, the construction of the Cambrian fan is quite different from the construction of the fan of clusters. In this talk I will define the Cambrian fan, discuss the connection to clusters and give a further connection to noncrossing partitions. All this will be illustrated with examples. (Many of these results are joint with David Speyer.)