Some Quotients of the Boolean Lattice are Symmetric Chain Orders Open Access

McKibben-Sanders, Jeremy (2010)

Permanent URL: https://etd.library.emory.edu/concern/etds/cn69m460x?locale=en
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Abstract

Abstract
Some Quotients of the Boolean Lattice are Symmetric Chain Orders
by Jeremy McKibben-Sanders


R. Canfield has conjectured that for all subgroups G of the symmetric group
Sn, the quotient Bn/G of the boolean lattice Bn is a symmetric chain order.
We provide a straightforward proof of K. K. Jordan's result that Bn/G is a
symmetric chain order when G is generated by an n-cycle, and we present a
simple algorithm for finding a symmetric chain decomposition of Bn/G, begin-
ning from the well-known symmetric chain decomposition of Bn obtained by
Greene and Kleitman. We also verify Canfield's conjecture when G is generated
by a set of pairwise disjoint transpositions, and provide an algorithm for finding
a symmetric chain decomposition of Bn/G in this case as well.

Table of Contents

1 Introduction
1.1 Basic Terminology
1.2 Symmetric Chain Decompositions
1.3 Permutation groups
1.4 The Main Result
1.5 Other Results
2 An Important SCD of Bn
3 Proof of the Main Result
4 Proof of Other Results
5 Closing Remarks

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