The Topology of the Unitary Dual of Crystallography Groups

TL;DR

This paper introduces a method to generate and analyze the topology of irreducible representations of crystallography groups across dimensions, demonstrated through detailed examples in three dimensions.

Contribution

It presents a novel procedure for generating irreducible representations and a strategy for exploring the topology of the unitary dual of crystallography groups.

Findings

Successfully generated all irreducible representations of a 3D crystallography group

Developed a sequence-based strategy to investigate the topology of the unitary dual

Provided explicit calculations as proof of concept

Abstract

We provide a procedure for generating the irreducible representations of crystallography groups in any dimension. We also furnish a strategy to investigate the topology of the unitary dual of a crystallography group using sequences of matrices. All irreducible representations (up to unitary equivalence) of the dimension 3 crystallography group 90 and some calculations involving sequences of these irreducible representations are included as a proof of concept of this procedure and strategy.