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 be a crystallographic reflection group with length function . We give a short and elementary derivation of the identity , where the product ranges over positive roots , and denotes the sum of the coordinates of with respect to the simple roots. We also prove that in the noncrystallographic case, this identity is valid in the limit ; i.e., .
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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
Article copyright: © Copyright 1998 American Mathematical Society