Aspects of Abstract Regular Polytopes and The Combinatorics of Coxeter Groups

Abstract

In this thesis, we examine abstract regular polytopes and some combinatorics of Coxeter groups. For abstract regular polytopes, we define the notion of when such polytopes are unravelled. We then go on to examine and catalogue examples of these abstract regular polytopes. We construct four different non-trivial infinite families and analyse some small interesting examples. Chapter 2 gives an introduction, some concrete examples and a bird's-eye-view of the existence of such polytopes before Chapters 3 and 4 construct the specific non-trivial families. In Chapter 5 we move on to Coxeter groups. Here we examine a neat combinatorial bijection between classes of reduced words of Coxeter groups and certain tilings of polygons known as Elnitsky's tilings. Chapter 6 examines the Bruhat order in relation to Elnitsky's tilings. In Chapter 7 we define E-embeddings; embeddings of Coxeter groups into the symmetric group that we show also give rise to bijections between tilings and reduced words. Chapter 8 provides an outline for a strategy to create E-embeddings but does not deliver an actual proof that this strategy indeed works. Chapter 9 examines the notions of `subtilings' of tilings in the context of our E-embeddings. Chapter 10 provides some suggestions for further research.

Date of Award	31 Dec 2022
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Peter Rowley (Supervisor) & Yuri Bazlov (Supervisor)