Subspaces of L^2(R^n) Invariant Under Crystallographic Shifts

Abstract

In 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.