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The following seems to be true: if $|W_q| := \sum {q^{l(w)}}$, where the sum is taken over the elements $w$, then $|W_q| = \prod {(1 + q +...+ q^{e_i})}$, where the product is taken over the exponents $e_i$.

In other words, if $V$ is the root space for $W$ and the polynomials $f_k$ are the basis of $S[V]$ sur $S^W[V]$, then the set of $deg(f_k)$ is the same with $l(w)$.

I couldn't find any proof for this. Yes, I tried Bourbaki.

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  • $\begingroup$ Yes this is true. Unfortunately I can't remember a proof or a good reference at the moment. (I believe the fact because the covariant ring is the cohomology ring of the flag variety which has a basis of Schubert classes.) $\endgroup$
    – Alexander Woo
    Commented Dec 14, 2012 at 5:20
  • $\begingroup$ Ty the expository article by Charles W. Curtis in the proceedings of an Ofxofrd Symposium ~ 1968- "Finite simple groups" edited by M.B. Powell and G.Higman. It is quite likely that Curtis has written elsewhere about such things,but that's teh reference that comes to mind. $\endgroup$
    – Geoff Robinson
    Commented Dec 14, 2012 at 5:47
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    $\begingroup$ @George: Note that the length of an element (relative to simple reflections) is unrelated to the order. Also, the results on Poincare series hold for all finite Coxeter groups, not just Weyl groups, and are treated in Bourbaki, V.5-V.6. The history is old and complicated, as I outlined in notes to my Chapter 3: Chevalley, Coxeter, etc. $\endgroup$
    – Jim Humphreys
    Commented Dec 14, 2012 at 14:36
  • $\begingroup$ Jim - yes, I meant "length", not "order", thanks for the correction. $\endgroup$
    – George
    Commented Dec 14, 2012 at 18:20

2 Answers 2

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I would leave this as a comment but I don't appear to have enough reputation points for that. Just to add to Philippe's answer that you will also find this as Theorem 10.2.3 in Carter's "Simple Groups of Lie Type", (it appears even earlier than this in Steinberg's Lecture Notes on Chevalley Groups, see Theorem 26 - pg. 131). Indeed this formula is key to giving factorisations of the orders of finite reductive groups as polynomials in $q$, (see Carter's "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters" - Section 2.9).

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The formula holds for any finite Coxeter group. A uniform proof can be found in Humphreys' Reflection groups and Coxeter groups, Section 3.15. A case by case proof is sketched in Björner and Brenti's Combinatorics of Coxeter groups, Section 7.1.

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