On the Poincaré series and cardinalities of finite reflection groups
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- by John R. Stembridge PDF
- Proc. Amer. Math. Soc. 126 (1998), 3177-3181 Request permission
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., $|W|=\prod (\operatorname {ht} (\alpha )+1)/\operatorname {ht}(\alpha )$.References
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Additional Information
- John R. Stembridge
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1109
- 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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3177-3181
- MSC (1991): Primary 20H15, 20F55
- DOI: https://doi.org/10.1090/S0002-9939-98-04473-6
- MathSciNet review: 1459151