Symmetric Chain Decompositions of Quotients of Chain Products by Wreath Products

TL;DR

This paper investigates conditions under which quotients of powers of chains by certain group actions, including wreath products, form symmetric chain decompositions, expanding the class of known symmetric chain orders.

Contribution

It establishes symmetric chain decompositions for quotients of chain products by specific groups, including wreath products, and provides elementary proofs for cyclic group actions on SCOs.

Findings

Quotients of chain powers by certain groups are symmetric chain orders.

Wreath product group actions preserve symmetric chain decompositions.

Cyclic group quotients of SCOs are also SCOs.

Abstract

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 $B_n/G$ 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.