Combinatorics of Root Systems

University Of Miami, Coral Gables FL

Investigators

Abstract

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.