Deformation of Chains via a Local Symmetric Group Action

A symmetric group action on the maximal chains in a finite, ranked poset is local if the adjacent transpositions act in such a way that (i, i+ 1) sends each maximal chain either to itself or to one differing only at rank i. We prove that when Sn acts locally on a lattice, each orbit considered as a subposet is a product of chains. We also show that all posets with local actions induced by labellings known as R∗S-labellings have symmetric chain decompositions and provide RS-labellings for the type B and D noncrossing partition lattices, answering a question of Stanley.