Abstract

In this thesis we study simultaneous realizations of multiple irreducible representations spaces within matrix algebras. In so doing we show how relations between irreducible representation spaces arise as a consequence of expressing fundamental and adjoint representation spaces as linearly independent subspaces. Our work proceeds in two parts. In both cases we work with the algebra M(8,C), which spans the space of eight by eight complex matrices. This space is chosen as it is: the smallest possible space to simultaneously incorporate the different gauge representations of the Standard Model, isomorphic to the complex Clifford algebra Cl(6), and a realization of the linear maps on the complexified Octonions. In the first part we present an explicit embedding of the Standard Model gauge groups. Second, we show the induction of a direct sum decomposition of the matrix algebra into a set of irreducible representation spaces. We discuss the features of and relationships between the irreducible representation spaces in the matrix algebra, and compare our results to features of Supersymmetry, Grand Unified Theories, and Noncommutative Geometry. Our work is not intended to be a derivation or explanation of Standard Model gauge representations. Instead, our work proposes a novel approach to studying combinations of irreducible representation spaces. As such this work explores the introduction of linear independence between irreducible representation spaces, the implications of this additional structure as realized in finite dimensional vector spaces, and relates our results to the Standard Model's irreducible representation spaces.