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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an infinite irreducible Coxeter group with the simple generators. We give a short proof that the word is reduced for any number of repetitions of . 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 which is obtained from the semi-infinite word by interchanging commuting elements and taking a prefix.

**1.**Anders Björner and Francesco Brenti,*Combinatorics of Coxeter groups*, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR**2133266****2.**Sergey Fomin and Andrei Zelevinsky,*Cluster algebras. IV. Coefficients*, Compos. Math.**143**(2007), no. 1, 112–164. MR**2295199**, 10.1112/S0010437X06002521**3.**C. Hohlweg, C. Lange and H. Thomas,*Permutahedra and Generalized Associahedra*,`arXiv:0709.4241`**4.**Robert B. Howlett,*Coxeter groups and 𝑀-matrices*, Bull. London Math. Soc.**14**(1982), no. 2, 137–141. MR**647197**, 10.1112/blms/14.2.137**5.**Mark Kleiner and Allen Pelley,*Admissible sequences, preprojective representations of quivers, and reduced words in the Weyl group of a Kac-Moody algebra*, Int. Math. Res. Not. IMRN**4**(2007), Art. ID rnm013, 28. MR**2338197**, 10.1093/imrn/rnm013**6.**Mark Kleiner and Helene R. Tyler,*Admissible sequences and the preprojective component of a quiver*, Adv. Math.**192**(2005), no. 2, 376–402. MR**2128704**, 10.1016/j.aim.2004.04.006**7.**Allen Knutson and Ezra Miller,*Subword complexes in Coxeter groups*, Adv. Math.**184**(2004), no. 1, 161–176. MR**2047852**, 10.1016/S0001-8708(03)00142-7**8.**D. Krammer,*The conjugacy problem for Coxeter groups*, Ph.D. thesis, Universiteit Utrecht, 1994. Available at`http://www.warwick.ac.uk/masbal/`**9.**Atsuo Kuniba, Kailash C. Misra, Masato Okado, Taichiro Takagi, and Jun Uchiyama,*Crystals for Demazure modules of classical affine Lie algebras*, J. Algebra**208**(1998), no. 1, 185–215. MR**1643999**, 10.1006/jabr.1998.7503**10.**Nathan Reading,*Cambrian lattices*, Adv. Math.**205**(2006), no. 2, 313–353. MR**2258260**, 10.1016/j.aim.2005.07.010**11.**Nathan Reading,*Sortable elements and Cambrian lattices*, Algebra Universalis**56**(2007), no. 3-4, 411–437. MR**2318219**, 10.1007/s00012-007-2009-1**12.**N. Reading and D. Speyer,*Cambrian Fans*, JEMS to appear,`arXiv:math.CO/0606201`.**13.**N. Reading and D. Speyer,*Sortable elements in infinite Coxeter groups*,`arXiv:0803.2722`.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
20F55

Retrieve articles in all journals with MSC (2000): 20F55

Additional Information

**David E. Speyer**

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

Email:
speyer@math.mit.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09638-X

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.