Palindromes and Orderings in Artin Groups

The braid group Bn, endowed with Artin’s presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn → Bn, v 7→ v̄, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x 7→ ∆x∆ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin groups (equipped with Artin’s presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We classify palindromes and palindromes invariant under τ in Artin groups of finite type. The tools are elementary rewriting and the construction of explicit left-orderings compatible with rev. Finally, we discuss generalizations to Artin groups of infinite type and Garside groups.