Realizations of Abstract Regular Polyhedra: Representation Theoretic and Geometric Approaches

Date of Award

8-2021

Document Type

Thesis

Degree Name

Master of Science in Mathematics

First Advisor

Mark L. Loyola, PhD

Abstract

An abstract regular polyhedron P is the combinatorial analog of a classical geometric polyhedron. It is a partially ordered set of elements called faces that can be completely characterized by a string C-group (G, T). A realization R is a mapping from P to a Eu- clidean G-space that is compatible with the associated orthogonal representation of G. This work expounds on two approaches to realizations – representation theoretic and geometric. In the first approach, we discuss the fundamentals of the theory of realizations culmi- nating in the decomposition of a realization into subrealizations akin to the decomposition of a real representation into its irreducible components. We adapt various computational tools and methods from linear algebra and the representation theory of finite groups to decompose R into its subrealizations. We also derive analogous results for several linear spaces arising naturally from the process of decomposition.

In the second approach, we put emphasis on the inherent geometric aspect of P. We accomplish this by formulating the notion of a geometric realization and identifying P with its image. This image, called a geometric polyhedron, is a 3-dimensional figure bounded by open sets of varying dimensions in space. These sets are suitably chosen based on their stabilizers and boundaries using a procedure which we adapted from the method of Wythoff construction. This procedure, which can now be directly applied to a string C-group (G, T), entails computing recursively the image of a real orthogonal representation of G of degree 3 applied to the said open sets.

To illustrate the concepts discussed and methods employed in both approaches, we consider the abstract regular polyhedra with automorphism group abstractly isomorphic to the noncrystallographic Coxeter group H3. In particular, we demonstrate how to construct the realizations of these polyhedra and how to decompose them into subrealizations. In the process, we are able to derive their pure realizations, completing the gap in the literature. On the other hand, using the adapted Wythoff construction, we reproduce the classical convex and star polyhedra with icosahedral symmetry, as well as construct non-standard icosahedral polyhedra with minimal surfaces as facets.

Key Words: abstract regular polyhedra, geometric polyhedra, realizations, string C-groups, orthogonal representations, Coxeter group H3

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