The Combinatorics of Coxeter Groups

Donald Coxeter was one of the most admirable modern geometers; applications of his work have been groundbreaking in the color-coding of virus proteins in chemistry, neutrino oscillation in physics, data mining, and even AIDS research. In this project, I consider the Coxeter groups, a well-studied and quite varied class of groups which have a remarkably diverse range of applications within mathematics and in the sciences where mathematics is applied. Subsets of Coxeter groups correspond to geometric regions which are tessellated (tiled) by so-called chambers; an example is the tessellation of the plane by equilateral triangles. The length function on the Coxeter group records algebraic information about each element (minimal length of an expression in terms of the generators) and geometric information about the corresponding chamber (distance from the fundamental chamber). Moreover, Poincaré series (or growth series) provide a way of enumerating the number of elements of each length in subsets of the group, or corresponding regions. I will describe some of the beautiful explicit formulae for the Poincaré series of the whole group, examine analogies between such series and others arising in combinatorics, and consider techniques for computing the Poincaré series of more general subsets. A remarkable feature of the subject is that the Poincaré series of the Coxeter groups, and many geometrically or algebraically natural subsets, are rational, implying that the number of elements of each length in these subsets can be efficiently enumerated.