Decomposition of direct product at an arbitrary Brillouin zone point: $D^{(\bigstar{R})(m)}$ $\otimes$ $D^{(\bigstar{-R})(m)}$

TL;DR

This paper presents a universal rule for decomposing the direct product of irreducible representations at any Brillouin zone point with its negative, applicable to all space groups and relevant for understanding double excitations in optical experiments.

Contribution

It introduces a general rule for decomposing the direct product of irreducible representations at arbitrary Brillouin zone points with their negatives, applicable across all space groups.

Findings

The number of zone center representation appearances equals the representation's dimensionality.

Double excitations from any Brillouin point have the symmetry to participate in optical experiments.

The rule is useful even when physics occurs away from high symmetry points.

Abstract

A general rule is presented for the decomposition of the direct product of irreducible representation at arbitrary Brillouin zone point $\bf{R}$ with its negative: the number of the appearences of the zone center representation equals the dimensionality of the representation. This rule is applicable for all space groups. Although in most situations the interesting physics takes place at high symmetry points in the Brillouin zone, this general rule is useful for situations where double excitations are considered. It is shown that double excitations from arbitrary Brillouin point $\bf{R}$ have the right symmetry to participate in all optical experiments regardless of polarization directions.