Symmetric Chain Decompositions of Quotients by Wreath Products

Abstract

Subgroups of the symmetric group $S_n$ act on $C^n$ (the $n$-fold product $C \times \cdots \times C$ of a chain $C$) by permuting coordinates, and induce automorphisms of the power $C^n$. For certain families of subgroups of $S_n$, the quotients defined by these groups can be shown to have symmetric chain decompositions (SCDs). These SCDs allow us to enlarge the collection of subgroups $G$ of $S_n$ for which the quotient $\mathbf{2}^n/G$ on the Boolean lattice $\mathbf{2}^n$ is a symmetric chain order (SCO). The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.