The PI proposes to study combinatorial aspects of root systems and reflection groups. This subject has been gaining importance, and the recent book of Bj""orner and Brenti (Combinatorics of Coxeter groups, Springer 2005) suggests that it has reached critical mass. In particular, there is a new subject of "Catalan phenomena" in Coxeter groups, which has coalesced from three topics: Garside structures and classifying spaces for braid groups (in terms of "noncrossing partitions"); the "cluster algebras" of Fomin and Zelevinsky; and the combinatorics of "diagonal harmonics", which arose from conjectures of Garsia and Haiman. The broad goal of the proposal is to explore and develop connections between these areas.
This research occurs in the field of "algebraic combinatorics". "Combinatorics" is the science of counting, arranging, and analyzing discrete structures. Historically, these discrete structures have come from computers --- from hardware, software, and computer networks --- hence the field of combinatorics has really emerged as a central area of mathematics in the last fifty years. More recently, the increasing speed of computers has allowed the use of combinatorial techniques in many areas of science --- in particular, in the analysis of genetic data and DNA structure. "Algebraic" combinatorics seeks to enrich the subject by incorporating ideas from more abstract, algebraic areas of mathematics. One such example is the "braid group", which has led to a popular new method of cryptography.
The Catalan numbers are some of the most important numbers in enumerative combinatorics. This project sought to bring some organization and structure to the subject of Catalan numbers via root systems and the theory of reflection groups. From the point of view of reflection groups there are three kinds of Catalan phenomena: 1. nonnesting phenomena, 2. noncrossing phenomena, and 3. cluster phenomena. It is still an open problem to fully explain the relationships between these three kinds. In particular, it is still an open problem to explain the occurrence of uniform enumerative formulas for noncrossing phenomena. This project made partial progress toward these goals. This project also led to new directions in the Catalan combinatorics of root systems. Specifically: The definition of the Ish arrangement of hyperplanes, in analogy with the Shi arrangement of hyperplanes. These are arrangements are related to the q,t-Catalan numbers. Also, a uniform bijection between noncrossing and nonnesting partitions for root systems. Also, a definition of parking functions (called noncrossing parking functions) that works even for noncrystallographic root systems. More recently, these ideas have led in the direction of Rational Catalan Combinatorics of root systems, which can be viewed as the character theory of finite dimensional representations of rational Cherednik algebras.
