Tensor operators and constructing indecomposable representations of semidirect product groups

Abstract

Consider a semidirect product group G=H⋉V, where H is reductive and V is a vector group. Two irreps π1 and π2 of H can be “assembled” into a representation of G if it is possible to construct an indecomposable representation Π of G whose restriction to H is π1⊕π2. It is shown that this is equivalent to the existence of a tensor operator from π2 to π1 carrying a representation of H which is equivalent to a nontrivial quotient of the representation which defines the semidirect product. This provides a systematic method for deciding whether two irreps can be assembled, and, if so, in how many inequivalent ways. The method is applied in many of the standard examples that arise in physical questions.