Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Crystallographic Shifts

In this thesis we consider crystal groups in dimension $n$ and their natural unitary representation on $L^2(\mathbb{R}^n)$. We show that this representation is unitarily equivalent to a direct integral of factor representations, and use this to characterize the subspaces of $L^2(\mathbb{R}^n)$ invariant under crystal symmetry shifts. Finally, by giving an explicit unitary equivalence of the natural crystal group representation, we find the \textit{central decomposition} guaranteed by direct integral theory.