Braid Groups, Clusters, and Free Probability

This document grew out of the workshop Braid Groups, Clusters, and Free Probability, which was held at the American Institute of Mathematics in Palo Alto, on January 10--14, 2005. The organizers of the workshop were Jon McCammond, Alexandru Nica, and Vic Reiner.

What follows is a joint statment of the participants regarding important open problems and promising directions for future progress in the subject. For further information, including a list of participants and participant abstracts, see the AIM webpage www.aimath.org. Jon McCammond also maintains a webpage with resources related to these topics [mccammond:webpage].

The goal of this workshop was to bring together mathematicians from different backgrounds to discuss a central theme which has recently emerged in many different contexts. Given a finite Coxeter group $W$, define the corresponding Catalan number $$ Cat(W)=\prod_{i=1}^n \frac{h+e_i+1}{e_i+1}, $$ where $h$ is the Coxeter number, and $e_1,e_2,\ldots,e_n$ are the exponents of the group $W$ (for notation related to Coxeter groups and reflection groups, we refer to [humphries]). When $W$ is the symmetric group $A_{n-1}$, this is just the usual Catalan number $Cat(A_{n-1})=\frac{1}{n+1}\binom{2n}{n}$. As with the classical Catalan numbers, these type $W$ Catalan numbers have a wealth of combinatorics associated with them, and they have recently appeared independently in several different fields, including Garside structures for braid groups, cluster algebras, and free probability.

This Catalan combinatorics describes extensive and surprising enumerative correspondences between these subjects, which in most cases are still unexplained (the term ``numerology'' is often used). A common goal of the workshop participants is to understand these concurrences, and search for underlying theories which can explain the combinatorics.

This is an exciting, emerging subject with many fundamental questions yet to be solved. We present here a collection of important and interesting questions offered by the participants. Many problems have attributions. These are to the person who brought the problem to my attention, and are used for the purpose of facilitating communication. No attempt has been made to track down the original source. A more detailed history can likely be found by contacting the contributor of the problem.