On the Poincaré series and cardinalities of finite reflection groups

Abstract:

Let $W$ be a crystallographic reflection group with length function $\ell (\cdot )$. We give a short and elementary derivation of the identity $\sum _{w\in W}q^{\ell (w)}=\prod (1-q^{\operatorname {ht} (\alpha )+1})/(1-q^{\operatorname {ht}(\alpha )})$, where the product ranges over positive roots $\alpha$, and $\operatorname {ht} (\alpha )$ 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 $q\to 1$; i.e., $|W|=\prod (\operatorname {ht} (\alpha )+1)/\operatorname {ht}(\alpha )$.