Symmetric Chain Decompositions of Quotients of Chain Products by Wreath Products

Subgroups of the symmetric group $S_n$ act on powers of chains $C^n$ by permuting coordinates, and induce automorphisms of the ordered sets $C^n$. The quotients defined are candidates for symmetric chain decompositions. We establish this for some families of groups in order to enlarge the collection of subgroups $G$ of the symmetric group $S_n$ for which the quotient $B_n\mathrm{/}G$ obtained from the $G$-orbits on the Boolean lattice $B_n$ is a symmetric chain order. The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.