Potter, Tom2025-10-142025-10-142025-10-14https://hdl.handle.net/10222/85472In this thesis we consider crystal groups in dimension n and their natural unitary representation on L^2(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(R^n) invariant under crystal symmetry shifts. Finally, by giving an explicit unitary equivalence of the natural crystal group representation, we find the central decomposition guaranteed by direct integral theory.encrystallographic groupscrystal symmetry shiftscentral decompositiondirect integral theoryunitary representation