Symmetry projection, geometry and choice of the basis

Abstract

A geometrical point of view of symmetry adapted projection to irreducible subspaces is presented. The projection is applied in two stages. The first step consists in projecting over subspaces spanning irreducible representations (irreps) of the symmetry group, while the second projection is carried out over the irreps of a subgroup defined through a suitable group chain. It is shown that choosing different chains is equivalent to propose alternative bases (passive point of view), while changing the projected function corresponds to the active point of view where the vector to be projected is rotated. Because of the importance of choosing the appropriate basis, an approach based on the concept of invariant operators to obtain the basis for discrete groups is presented. We show that this approach is analogue to the case of continuum groups and it is closely related to the definition of quantum numbers. The importance of these concepts is illustrated through the effect of symmetry breaking. Because of the deep insight into the group theory concepts, we believe the presented viewpoint helps to understand the main ingredients involved in group representation theory using the latest advances in the subject for discrete groups.