Deformation of chains via a local symmetric group action
A symmetric group action on the maximal chains in a finite, ranked poset \(P\) was defined by \textit{R. P. Stanley} [Electron. J. Comb. 3, No. 2, Research paper R6 (1996; Zbl 0857.05091); printed version J. Comb. 3, No. 2, 161-182 (1996)] to be local if for each \(i\) the transposition \((i,i+1)\) sends each maximal chain either to itself or to one differing only at rank \(i\). A prototypical example is the natural action of \(S_n\) on the maximal chains of the Boolean algebra \({\mathcal B}_n\). Here the only orbit is \({\mathcal B}_n\) itself, which is a product of 2-chains. The main result of the paper under review says that this is related only with the fact that \({\mathcal B}_n\) is a lattice: each orbit under a local action of \(S_n\) on the maximal chains of a lattice is a product of chains. For a general poset \(P\) it is shown that the Frobenius characteristic of a local action of \(S_n\) on the maximal chains of \(P\) is an \(h\)-positive symmetric function. Some applications to symmetric chain decompositions of posets and to noncrossing partition lattices are also discussed.
scientific article; zbMATH DE number 1309970