Freely braided elements in Coxeter groups

Details Freely braided elements in Coxeter groups Freely braided elements in Coxeter groups

2003-01-10

arxiv.org/abs/math/0301104v2

Annals of Combinatorics 6 (2002), 337-348

Mathematics

Combinatorics

18 pages, AMSTeX. Results renumbered to agree with published version

Scientific paper

We introduce a notion of "freely braided element" for simply laced Coxeter groups. We show that an arbitrary group element $w$ has at most $2^{N(w)}$ 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 $w$.

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