Flag-symmetry of the poset of shuffles and a local action of the symmetric group Original Research Article

We show that the posets of shuffles introduced by Greene in 1988 are flag symmetric, and we describe a permutation action of the symmetric group on the maximal chains which is local and yields a representation of the symmetric group whose character has Frobenius characteristic closely related to the flag symmetric function. A key tool is provided by a new labeling of the maximal chains of a poset of shuffles. This labeling and the structure of the orbits of maximal chains under the local action lead to combinatorial derivations of enumerative properties obtained originally by Greene. As a further consequence, a natural notion of type of shuffles emerges and the monoid of multiplicative functions on the poset of shuffles is described in terms of operations on power series. The main results concerning the flag symmetric function and the local action on the maximal chains of a poset of shuffles are obtained from new general results regarding chain labelings of posets.