Date of Completion

5-4-2017

Embargo Period

5-4-2017

Keywords

combinatorics, group, homomesy, independent set, orbit, set partition

Major Advisor

Tom Roby

Associate Advisor

Ralf Schiffler

Associate Advisor

David Reed Solomon

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

This paper explores the orbit structure and homomesy properties of various actions on finite sets. The homomesy phenomenon, meaning constant averages over orbits, was proposed by Propp and Roby in 2011. For many of the known instances of homomesy, Reiner, Stanton, and White's cyclic sieving phenomenon (CSP) is also present. However, we prove homomesy for several maps whose order is large relative to the size of the set, implying that a natural CSP is unlikely. Sometimes we can prove facts about the orbit sizes either as a corollary to the homomesy or by the technique used to prove homomesy.

Many of the actions we describe are products of much simpler ones. Among these, we consider maps defined as products of simple "toggling" involutions. These come from the Striker's theory of generalized toggle groups, an active area of research in dynamical algebraic combinatorics. Several known instances of homomesy have been discovered for elements of toggle groups. While the individual toggles have order two, the order of a composition of several toggles is more difficult to analyze. We also consider an action of "whirling," due to Propp, that can be defined for any family of functions between finite sets. This action is also the composition of simpler ones.

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