Polynomial divided difference operators satisfying the braid relations

Many interesting families of polynomials are indexed by permutations or related objects, and are defined by applying divided difference operators, modified by polynomials, on some initial base case. The fact that these constructions produce well-defined polynomials is based on the applied modified divided difference operators satisfying the braid relations. We thus determine all the families of polynomial divided difference operators that satisfy the braid relations, complementing an old result of Lascoux and Schützenberger.