PALINDROMES AND ORDERINGS IN ARTIN GROUPS

The braid group B_n, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : B_n →B_n, , defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ^-1xΔ 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 study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.