On minimal lengths of expressions of Coxeter group elements as products of reflections
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Abstract:
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.References
- Hélène Barcelo and Alain Goupil, Combinatorial aspects of the Poincaré polynomial associated with a reflection group, Jerusalem combinatorics ’93, Contemp. Math., vol. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 21–44. MR 1310572, DOI 10.1090/conm/178/01890
- N. Bourbaki, Groupes ét algebres de Lie, Ch. 4–6, Hermann, Paris, 1964.
- Francesco Brenti, A combinatorial formula for Kazhdan-Lusztig polynomials, Invent. Math. 118 (1994), no. 2, 371–394. MR 1292116, DOI 10.1007/BF01231537
- R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 318337
- Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511. MR 782232, DOI 10.1007/BF01388520
- M. J. Dyer, Hecke algebras and reflections in Coxeter groups, Ph.D. thesis, Univ. of Sydney, 1987.
- Matthew Dyer, Reflection subgroups of Coxeter systems, J. Algebra 135 (1990), no. 1, 57–73. MR 1076077, DOI 10.1016/0021-8693(90)90149-I
- Matthew Dyer, On the “Bruhat graph” of a Coxeter system, Compositio Math. 78 (1991), no. 2, 185–191. MR 1104786
- M. J. Dyer, Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 141–165. MR 1197833, DOI 10.1090/conm/139/1197833
- M. J. Dyer, Hecke algebras and shellings of Bruhat intervals, Compositio Math. 89 (1993), no. 1, 91–115. MR 1248893
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- Louis Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57–64. MR 154929, DOI 10.1017/S0027763000011028
Additional Information
- Matthew J. Dyer
- Affiliation: Department of Mathematics, University of Notre Dame, Room 370 CCMB, Notre Dame, Indiana 46556-5683
- Email: Dyer.1@nd.edu
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2591-2595
- MSC (2000): Primary 20F55, 22E47, 06A07
- DOI: https://doi.org/10.1090/S0002-9939-01-05876-2
- MathSciNet review: 1838781