  The complex irreducible representations of the symmetric group carry an
important canonical basis called the Kazhdan-Lusztig basis. Although it is
difficult to express how general permutations act on this basis, some
distinguished permutations have beautiful descriptions. In 2010 Rhoades showed
that the long cycle $(1, 2,..., n)$ acts by the jeu-de-taquin promotion
operator in the case when the irreducible representation is indexed by a
rectangular partition. We prove a generalisation of this theorem in two
directions: on the one hand we lift the restriction on the shape of the
partition, and on the other hand we enlarge the result to the collection of all
separable permutations.
