Irreducible Representations of Diperiodic Groups

TL;DR

This paper calculates the irreducible representations of all 80 diperiodic groups, providing detailed tables and discussing their physical applications such as energy band degeneracy and selection rules.

Contribution

It systematically derives and tabulates the irreducible representations of all diperiodic groups, a comprehensive resource for symmetry analysis in two-dimensional periodic systems.

Findings

Complete tables of irreducible representations for 80 diperiodic groups.

Insights into degeneracy and topology of energy bands.

Application of group theory to physical properties of periodic systems.

Abstract

The irreducible representations of all of the 80 diperiodic groups, being the symmetries of the systems translationally periodical in two directions, are calculated. To this end, each of these groups is factorized as the product of a generalized translational group and an axial point group. The results are presented in the form of the tables, containing the matrices of the irreducible representations of the generators of the groups. General properties and some physical applications (degeneracy and topology of the energy bands, selection rules, etc.) are discussed.