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Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Shifts by a Crystal Group

Tom Potter, Keith Taylor

TL;DR

This work provides a complete characterization of $\\Gamma$-shift invariant closed subspaces of $L^2(\\mathbb{R}^n)$ for crystal groups $\\Gamma$ by translating the problem into a fibered range-function model. Through a sequence of unitary reductions, the natural representation is transformed to an equivalent fibered form on $\\Pi\\Omega\\times\\ell^2(\\mathcal{L}^*)$, yielding a one-to-one correspondence with measurable range functions valued in subspaces of $\\ell^2(\\mathcal{L}^*)$. The main condition ties Fourier samples $\\widehat{f}(L(\\omega+\\nu))$ to a fiber function via $\\widehat{f}(L(\\omega+\\nu))=e^{2\pi i\\nu\cdot x_L}\langle F(L\\omega),\\delta_\\nu\rangle$, and a twisted variant with $J^{\\Gamma}$ is also given. The results unify previous shift-invariant theory and provide a robust framework for analyzing band representations in topological quantum chemistry, including explicit non-symmorphic examples.

Abstract

For a crystal group $Γ$ in dimension $n$, a closed subspace $\mathcal{V}$ of $L^2(\mathbb{R}^n)$ is called $Γ$--shift invariant if, for every $f\in\mathcal{V}$, the shifts of $f$ by every element of $Γ$ also belong to $\mathcal{V}$. The main purpose of this paper is to provide a characterization of the $Γ$--shift invariant closed subspaces of $L^2(\mathbb{R}^n)$.

Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Shifts by a Crystal Group

TL;DR

This work provides a complete characterization of -shift invariant closed subspaces of for crystal groups by translating the problem into a fibered range-function model. Through a sequence of unitary reductions, the natural representation is transformed to an equivalent fibered form on , yielding a one-to-one correspondence with measurable range functions valued in subspaces of . The main condition ties Fourier samples to a fiber function via , and a twisted variant with is also given. The results unify previous shift-invariant theory and provide a robust framework for analyzing band representations in topological quantum chemistry, including explicit non-symmorphic examples.

Abstract

For a crystal group in dimension , a closed subspace of is called --shift invariant if, for every , the shifts of by every element of also belong to . The main purpose of this paper is to provide a characterization of the --shift invariant closed subspaces of .
Paper Structure (6 sections, 16 theorems, 41 equations, 3 figures)

This paper contains 6 sections, 16 theorems, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Crystal groups
  4. Several Unitary Transformations
  5. The natural representation
  6. Invariant closed subspaces

Key Result

Lemma 2.1

Let $\mathcal{L}$ be a lattice in $\mathbb{R}^n$ and let $\mathcal{L}^*$ be its dual lattice. Let $g\in L^1(\mathbb{R}^n)$. Then $\sum_{\nu\in\mathcal{L}^*}g(\theta+\nu)$, converges for a.e. $\theta\in\mathbb{R}^n$ and $\mathcal{P}_{\mathcal{L}^*}g|_Q\in L^1(Q)$.

Figures (3)

  • Figure 1: A pattern illustrating the symmetries of the wallpaper group $pg$.
  • Figure 2: The $\Gamma^*$-orbit of $\omega_0$ and domain $\Omega$.
  • Figure 3: The measurable subset $E$ of $\Omega$ and three of its shifts.

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 3.1
  • ...and 24 more