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Powers of Coxeter elements in infinite groups are reduced

Author: David E. Speyer
Journal: Proc. Amer. Math. Soc. 137 (2009), 1295-1302
MSC (2000): Primary 20F55
Published electronically: October 29, 2008
MathSciNet review: 2465651
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Abstract: Let $ W$ be an infinite irreducible Coxeter group with $ (s_1, \ldots, s_n)$ the simple generators. We give a short proof that the word $ s_1 s_2 \cdots s_n s_1 s_2 \cdots$ $ s_n \cdots s_1 s_2 \cdots s_n$ is reduced for any number of repetitions of $ s_1 s_2 \cdots s_n$. This result was proved for simply laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for $ w_0$ which is obtained from the semi-infinite word $ s_1 s_2 \cdots s_n s_1 s_2 \cdots s_n \cdots$ by interchanging commuting elements and taking a prefix.

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Additional Information

David E. Speyer
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139

Received by editor(s): February 11, 2008
Received by editor(s) in revised form: May 8, 2008
Published electronically: October 29, 2008
Communicated by: Jim Haglund
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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