Symmetric Chain Decompositions of Quotients by Wreath Products

Dwight Duffus,Kyle Thayer

doi:10.37236/5073

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.

Highlights

We are interested in a familiar symmetry property of finite partially ordered sets – possessing a partition into symmetric chains
For certain families of subgroups of Sn, 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 Sn for which the quotient 2n/G on the Boolean lattice 2n is a symmetric chain order (SCO)

Summary

Introduction

We are interested in a familiar symmetry property of finite partially ordered sets – possessing a partition into symmetric chains. We are concerned with a particular family of SCOs, products of finite chains, and whether quotients of these are SCOs. For any partially ordered set P and any subgroup G of Aut(P ) the automorphism group of P , let P/G denote the quotient poset. We [7] showed that 2n/G is an SCO provided that G is generated by powers of disjoint cycles This result generalizes Jordan’s result a bit, but the method of proof is likely more interesting in that we construct an SCD of the quotient by pruning the Greene-Kleitman SCD in a more direct way than Jordan. 2n/G is a symmetric chain order for any subgroup G of Aut(2n) defined as follows: G = K T , the wreath product of K by T , where K is a subgroup of Sk, T is a subgroup of St, and both K and T are generated by powers of disjoint cycles. They possess several symmetry properties but are not known to be SCOs

Proof of Theorem 1

The required properties

Powers of disjoint cycles suffice

Proof of Theorem 2

Abelian subgroups of Sn

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