Condition That the Irreducible Representations of a Group, Considered as Representations of a Subgroup, Do Not Contain Any Representation of the Subgroup More Than Once

Abstract

The interest in the condition contained in the title stems from the desirability to specify a definite form for all irreducible representations of a group, i.e. to specify a particular coordinate system, with some simple properties, in representation space. In order to do this, let us first decompose each representation space into orthogonal subspaces, each such subspace being the representation space of an irreducible representation of a subgroup. If these irreducible representations are all inequivalent, their subspaces are uniquely given and can be characterized by the corresponding irreducible representation of the subgroup. Once this is done, one can proceed in the same way with the subgroup and, if possible, continue until an abelian subgroup of the last group is reached. Hence, if there is a sequence of subgroups G₁, G₂,... of the original group G₀ such that G _{l +1} is a subgroup of G _l of the nature specified in the title, a coordinate axis in the space of an irreducible representation of G₀ can be specified by enumerating the irreducible representations of all the subgroups G₁, G₂,... to the representation spaces of which the vector belongs.