Representations of the symmetry groups of infinite crystals

TL;DR

This paper explores the complex representation theory of infinite crystal symmetry groups, employing harmonic analysis to overcome the limitations of traditional finite group methods, with applications to magnetic and non-magnetic crystals.

Contribution

It introduces a harmonic analysis approach to study representations of infinite crystal symmetry groups, extending beyond standard character theory and addressing magnetic groups in various dimensions.

Findings

Developed methods for decomposing representations into irreducible components.

Applied harmonic analysis to infinite non-compact groups of crystal symmetries.

Analyzed Mackey's restriction, tensor products, and symmetric/antisymmetric squares of induced representations.

Abstract

We investigate the representations of the symmetry groups of infinite crystals. Crystal symmetries are usually described as the finite symmetry group of a finite crystal with periodic boundary conditions, for which the Brillouin zone is a finite set of points. However, to deal with the continuous crystal momentum $\mathbf{k}$ required to discuss the continuity, singularity or analyticity of band energies $\epsilon_n(\mathbf{k})$ and Bloch states $\psi_{\mathbf{k}}$, we need to consider infinite crystals. The symmetry groups of infinite crystals belong to the category of infinite non-compact groups, for which many standard tools of group theory break down. For example, character theory is no longer available for these groups and we use harmonic analysis to build the group algebra, the regular representation, the induction of irreducible representations of the crystallographic group from projective representations of the point groups and the decomposition of a representation into its irreducible parts. We deal with magnetic and non-magnetic groups in arbitrary dimensions. In the last part of the paper, we discuss Mackey's restriction of an induced representation to a subgroup, the tensor product of induced representations and the symmetric and antisymmetric squares of induced representations.