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

TL;DR

This thesis characterizes subspaces of L^2(R^n) invariant under crystal group shifts by decomposing the natural representation into factor representations and explicitly finding the central decomposition.

Contribution

It provides a detailed analysis of the unitary representation of crystal groups on L^2(R^n) and explicitly constructs the central decomposition of this representation.

Findings

Unitary equivalence to a direct integral of factor representations

Characterization of invariant subspaces under crystal symmetry shifts

Explicit construction of the central decomposition

Abstract

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.