4 - Root Systems and Coxeter groups

Summary

Introduction and Overview

A fundamental aspect of the structure of orthogonal polynomials of classical type is the use of finite reflection groups. This chapter presents the part of the theory which is needed for analysis. We refer to the books by Coxeter [1973], Grove and Benson [1985] and Humphreys [1990] for the algebraic structure theorems. We will begin with the orthogonal groups, the definitions of reflections and root systems, and descriptions of the infinite families of finite reflection groups. A key part of the chapter is the definition and fundamental theorems for the differential–difference (Dunkl) operators.

For x, y ∈ ℝ^d the inner product is (x, y) = X∧j=i xjVj∧ a nd the norm is ||x|| = (a∧x)1/2. A matrix ω = {wij)d i-1 is called orthogonal if ωω^T = I_d (where ω^T denotes the transpose of ω and I_d is the d χ d identity matrix. Equivalent conditions for orthogonality are:

1. ω is invertible and ω⁻¹ = ω^T;

2. for each x G Rd, \\x\\ = \\xw\\;

3. for each x, y G Ed, (x, y) = (xw, yw);

4. the rows of ω form an orthonormal basis for ℝ^d.

The set of orthogonal matrices is closed under multiplication and inverses (by condition (ii), for example) and forms the orthogonal group, denoted O(d). Condition (iv) shows that O(d)) is a closed bounded subset of all d x d matrices, hence is a compact group. If w G O(d), then detω = ±1. The subgroup SO(d)) = {ω ∈ O(d) : det ω = 1} is called the special orthogonal group.