primitive degree of a coxeter group

Question

Please, can you explain to a none specialist what does mean ' primitive degree' of a coxeter group ?

Thanks in advance.

Answer 1

To clarify a bit what Igor refers to, the answer to the question asked is that "primitive degrees" for a finite Coxeter group such as a Weyl group are usually abbreviated to "degrees", meaning the uniquely determined degrees of basic homogeneous invariant polynomials of the group (whose product is the order of the group). Here the natural class of groups to consider, as seen in Chevalley's 1955 Amer. J. Math. paper, would be finite real or complex reflection groups. The real ones were classified by Coxeter, who also found the nice presentation which motivates the current name "Coxeter groups".

Opdam's paper deals more generally with complex reflection groups, classified by Shephard-Todd, only some of which are Coxeter groups but all of which are characterized by the fact that their algebra of invariant polynomials is itself a polynomial algebra on homogeneous generators of uniquely determined degrees.

Answer 2

Look at page 2 of:

http://arxiv.org/PS_cache/math/pdf/9808/9808026v1.pdf

(in case the link does not work: Opdam, Complex reflection groups and fake degrees)