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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the Poincaré series and cardinalities
of finite reflection groups


Author: John R. Stembridge
Journal: Proc. Amer. Math. Soc. 126 (1998), 3177-3181
MSC (1991): Primary 20H15, 20F55
MathSciNet review: 1459151
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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., $|W|=\prod (\operatorname{ht} (\alpha )+1)/\operatorname{ht}(\alpha )$.


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Additional Information

John R. Stembridge
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1109

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04473-6
PII: S 0002-9939(98)04473-6
Received by editor(s): October 9, 1996
Received by editor(s) in revised form: March 29, 1997
Additional Notes: The author was partially supported by a grant from the NSF
Communicated by: Jeffry N. Kahn
Article copyright: © Copyright 1998 American Mathematical Society