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On minimal lengths of expressions of Coxeter group elements as products of reflections

Author: Matthew J. Dyer
Journal: Proc. Amer. Math. Soc. 129 (2001), 2591-2595
MSC (2000): Primary 20F55, 22E47, 06A07
Published electronically: February 9, 2001
MathSciNet review: 1838781
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Abstract | References | Similar Articles | Additional Information


It is shown that the absolute length $l'(w)$of a Coxeter group element $w$ (i.e. the minimal length of an expression of $w$ as a product of reflections) is equal to the minimal number of simple reflections that must be deleted from a fixed reduced expression of $w$ so that the resulting product is equal to $e$, the identity element. Also, $l'(w)$ is the minimal length of a path in the (directed) Bruhat graph from the identity element $e$ to $w$, and $l'(w)$ is determined by the polynomial $R_{e,w}$ of Kazhdan and Lusztig.

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  • 1. H. Barcelo and A. Goupil, Combinatorial aspects of the Poincaré polynomial associated with a reflection group, Jerusalem Combinatorics '93 (Providence, R.I.), Contemp. Math., vol. 178, Amer. Math. Soc, 1994, pp. 21-44. MR 96e:20061
  • 2. N. Bourbaki, Groupes ét algebres de Lie, Ch. 4-6, Hermann, Paris, 1964.
  • 3. F. Brenti, A combinatorial formula for Kazhdan-Lusztig polynomials, Invent. Math 118 (1994), 371-394. MR 96c:20074
  • 4. R. W. Carter, Conjugacy classes in the Weyl group, Comp. Math. 25 (1972), 1-59. MR 47:6884
  • 5. V. V. Deodhar, On some geometric aspects of Bruhat orderings. I. a finer decomposition of Bruhat cells, Invent. Math. 79 (1985), 499-511. MR 86f:20045
  • 6. M. J. Dyer, Hecke algebras and reflections in Coxeter groups, Ph.D. thesis, Univ. of Sydney, 1987.
  • 7. -, Reflection subgroups of Coxeter systems, J. of Alg. 135 (1990), 57-73. MR 91j:20100
  • 8. -, On the ``Bruhat graph'' of a Coxeter system, Comp. Math. 78 (1991), 185-191. MR 92c:20076
  • 9. -, Hecke algebras and shellings of Bruhat intervals II: twisted Bruhat orders, Kazhdan-Lusztig theory and related topics (V. V. Deodhar, ed.), Contemp. Math., vol. 139, 1992, pp. 141-165. MR 94c:20072
  • 10. -, Hecke algebras and shellings of Bruhat intervals, Comp. Math. 89 (1993), 91-115. MR 95c:20053
  • 11. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, no. 29, Camb. Univ. Press, Cambridge, 1990. MR 92h:20002
  • 12. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • 13. L. Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57-64. MR 27:4872

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Additional Information

Matthew J. Dyer
Affiliation: Department of Mathematics, University of Notre Dame, Room 370 CCMB, Notre Dame, Indiana 46556-5683

Received by editor(s): August 23, 1999
Received by editor(s) in revised form: January 27, 2000
Published electronically: February 9, 2001
Communicated by: John R. Stembridge
Article copyright: © Copyright 2001 American Mathematical Society

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