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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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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  • James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
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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