On the Poincar\'e series and cardinalities of finite reflection groups
Proc. Amer. Math. Soc. 126 (1998) 3177-3181.
John R. Stembridge
Department of Mathematics, University of Michigan, Ann Arbor,
Michigan 48109-1109
Abstract. Let $W$ be a crystallographic reflection group with length function
$\ell(.)$. We give a short and elementary derivation of the identity
$\sum_{w\in W} q^{\ell(w)} = \prod(1-q^{{\rm ht}(\alpha)+1})/
(1-q^{{\rm ht}(\alpha)})$, where the product ranges over positive roots
$\alpha$, and ${\rm ht}(\alpha)$ denotes the sum of the coordinates of
$\alpha$ with respect to the simple roots. We also prove that in the
noncrystallographic case, this identity is valid in the limit
$q \rightarrow 1$; i.e., $|W|=\prod({\rm ht}(\alpha)+1)/{\rm ht}(\alpha)$.