Some Quotients of the Boolean Lattice are Symmetric Chain Orders

TL;DR

The paper proves that certain quotients of the Boolean lattice, specifically those generated by powers of disjoint cycles, are symmetric chain orders, extending previous results and utilizing Greene and Kleitman's decompositions.

Contribution

It provides a straightforward proof that quotients of the Boolean lattice by groups generated by powers of disjoint cycles are symmetric chain orders.

Findings

B(n)/G is an SCO when G is generated by powers of disjoint cycles

Utilizes Greene and Kleitman's symmetric chain decompositions

Extends previous results on Boolean lattice quotients

Abstract

R. Canfield has conjectured that for all subgroups G of the automorphism group of the Boolean lattice B(n) (which can be regarded as the symmetric group S(n)) the quotient order B(n)/G is a symmetric chain order. We provide a straightforward proof of a generalization of a result of K. K. Jordan: namely, B(n)/G is an SCO whenever G is generated by powers of disjoint cycles. The symmetric chain decompositions of Greene and Kleitman provide the basis for partitions of these quotients.