8 - Subduced and induced representations

Summary

Let G be a finite group and H be a subgroup of G. Then a representation of the group G automatically describes a representation of the subgroup H of G. Such a representation is called a representation of H subduced by a representation of G. Conversely, from a given representation of a subgroup H of G we can form a representation of the group G. Such a representation is called a representation of G induced by a representation of its subgroup H. The problem is to form the irreducible representations (irreps in short) of G from the irreps of its subgroup H. If the group G is finite and solvable (see Section 8.4.1), the problem of forming the irreps of G may be solved by a step-by-step procedure from the trivial irrep of the trivial identity subgroup. This method is possible, for example, for a crystallographic point group. An alternative approach is via the induced irreps of G from the so-called small representations of the little groups of the irreps of H. As a preparation, we shall discuss subduced representations first.

Subduced representations

Let G = {g} be a group and H = {h} be a subgroup of G. Let Γ(G) = {Γ(g); g ∈ G} be a representation of G, then it provides a representation of H by {Γ(h); h ∈ H}. This representation is called the subduced representation of Γ(G) onto H or the representation of H subduced by the representation Γ(G).