If W is a finite subgroup of GL(V) generated by reflections, it acts in a natural way on the ring of polynomial functions on V. This chapter will be devoted to the study of this action, emphasizing the remarkable features of the subring of invariants, which turns out to be a polynomial ring on generators of certain well-determined degrees (whose product is |W|). This is a far-reaching generalization of the fundamental theorem on symmetric polynomials (the case of a symmetric group).

After some generalities on invariants of arbitrary finite groups (3.1) – (3.2), we prove the fundamental theorem of Chevalley giving an algebraically independent set of generators for the ring of invariants (3.3) – (3.5) and observe (3.7) that their degrees are uniquely defined. Moreover, the sum and product of the degrees have natural interpretations (3.9). A standard Jacobian criterion for algebraic independence of polynomials (3.10) allows us to work out some examples (3.12). The degrees enter in a surprising way into the factorization of the Poincaré polynomial of W (3.15).

In 3.16–3.19 we find a completely different interpretation of the degrees, in terms of the eigenvalues of a ‘Coxeter element’ of W (the product of simple reflections in some order). For Weyl groups the calculation of degrees can also be done by counting roots of each height (3.20).

Polynomial invariants of a finite group

Before dealing specifically with reflection groups, let us consider what can be said about the polynomial invariants of an arbitrary finite subgroup of GL(V), where V is an n-dimensional vector space over a field K of characteristic 0.