Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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 $ 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.

References [Enhancements On Off] (What's this?)

  • 1. A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer-Verlag, 2005. MR 2133266 (2006d:05001)
  • 2. S. Fomin and A. Zelevinsky, Cluster algebras IV, Coefficients, Compositio Mathematica 143 (2007), 112-164. MR 2295199 (2008d:16049)
  • 3. C. Hohlweg, C. Lange and H. Thomas, Permutahedra and Generalized Associahedra, arXiv:0709.4241
  • 4. R. B. Howlett, Coxeter groups and $ M$-matrices, Bulletin of the London Mathematical Society 14 (1982), no. 2, 137-141. MR 647197 (83g:20032)
  • 5. M. Kleiner and A. Pelley, Admissible sequences, preprojective representations of quivers, and reduced words in the Weyl group of a Kac-Moody algebra, International Mathematics Research Notices (2007), no. 4, Art. ID mm013. MR 2338197 (2008f:16035)
  • 6. M. Kleiner and H. R. Tyler, Admissible sequences and the preprojective component of a quiver, Advances in Mathematics 192 (2005), no. 2, 376-402. MR 2128704 (2006d:16026)
  • 7. A. Knutson and E. Miller, Subword complexes in Coxeter groups, Advances in Mathematics 184 (2004), no. 1, 161-176. MR 2047852 (2005c:20066)
  • 8. D. Krammer, The conjugacy problem for Coxeter groups, Ph.D. thesis, Universiteit Utrecht, 1994. Available at$ \sim$masbal/
  • 9. A. Kuniba, K. Misra, M. Okado, T. Takagi, and J. Uchiyama, Crystals for Demazure modules of classical affine Lie algebras, Journal of Algebra 208 (1998), no. 1, 185-215. MR 1643999 (99h:17008)
  • 10. N. Reading, Cambrian Lattices, Advances in Mathematics 205 (2006), no. 2, 313-353. MR 2258260 (2007g:05195)
  • 11. N. Reading, Sortable elements and Cambrian lattices, Algebra Universalis 56 (2007), no. 3-4, 411-437. MR 2318219 (2008d:20073)
  • 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.

Similar Articles

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

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.

American Mathematical Society