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The dual braid monoid. (English) Zbl 1064.20039
Summary: We study a new monoid structure for Artin groups associated with finite Coxeter systems. Like the classical positive braid monoid, the new monoid is a Garside monoid. We give several equivalent constructions: algebraically, the new monoid arises when studying Coxeter systems in a “dual” way, replacing the pair \((W,S)\) by \((W,T)\), with \(T\) the set of all reflections; geometrically, it arises when looking at the reflection arrangement from a certain basepoint. In the type \(A\) case, we recover the monoid constructed by Birman, Ko and Lee.

MSC:
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
20M05 Free semigroups, generators and relations, word problems
20F55 Reflection and Coxeter groups (group-theoretic aspects)
57M07 Topological methods in group theory
Software:
GAP
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