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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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


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., $|W|=\prod (\operatorname {ht} (\alpha )+1)/\operatorname {ht}(\alpha )$.
  • R. Beerends, “On the Abel Transform and its Inversion,” Ph. D. thesis, University of Leiden, 1987.
  • Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
  • 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
  • I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161–174. MR 322069, DOI 10.1007/BF01431421
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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:
  • MathSciNet review: 1459151