Palindromes and orderings in Artin groups

Abstract

The braid group $B_{n}$, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism ${\rm{rev}}: B_{n} \to B_{n}$, $v \mapsto \bar{v}$, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation $\tau:x \mapsto \Delta^{-1}x \Delta$ by the generalized half-twist (Garside element). More generally, the involution ${\rm{rev}}$ is defined for all Artin groups (equipped with Artin's presentation) and the involution $\tau$ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We classify palindromes and palindromes invariant under $\tau$ 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.