Freely Braided Elements in Coxeter Groups
Copyright policy )Freely Braided Elements in Coxeter Groups
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 $w$.
18 pages, AMSTeX. Results renumbered to agree with published version
- Long Island University United States
- Lancaster University United Kingdom
Generators, relations, and presentations of groups, Free semigroups, generators and relations, word problems, root systems, root sequences, commutation graphs, Group Theory (math.GR), Graphs and abstract algebra (groups, rings, fields, etc.), Reflection and Coxeter groups (group-theoretic aspects), commutation classes, FOS: Mathematics, simply laced Coxeter groups, Mathematics - Combinatorics, Combinatorics (math.CO), 20F55, Mathematics - Group Theory, braid relations
Generators, relations, and presentations of groups, Free semigroups, generators and relations, word problems, root systems, root sequences, commutation graphs, Group Theory (math.GR), Graphs and abstract algebra (groups, rings, fields, etc.), Reflection and Coxeter groups (group-theoretic aspects), commutation classes, FOS: Mathematics, simply laced Coxeter groups, Mathematics - Combinatorics, Combinatorics (math.CO), 20F55, Mathematics - Group Theory, braid relations
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