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Freely Braided Elements in Coxeter Groups
R.M. Green1 and J. Losonczy2
1Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, England
r.m.green@lancaster.ac.uk
2Department of Mathematics, Long Island University, Brookville, NY 11548, USA
Losonczy@liu.edu
Annals of Combinatorics 6 (3) p.337-348 September, 2002
AMS Subject Classification: 20F55
Abstract:
We introduce a notion of ``freely braided element" for simply laced Coxeter groups. We show that an arbitrary group element w has at most 2N(w) commutation classes of reduced expressions, where N(w) is a certain statistic defined in terms of the positive roots made negative by w. This bound is achieved if w is freely braided. In the type A setting, we show that the bound is achieved only for freely braided w.
Keywords: braid relation, commutation class, Coxeter group, root sequence

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