On the irreducible representations of groups containing a subgroup of finite index

Abstract

The objects under consideration are: A group G containing a subgroup S of finite index p, an irreducible representation (= multiplier representation by unitary or by unitary and antiunitary operators on a Hilbert space of arbitrary dimension) U of G, and an irreducible representation W of S. It is shown (1) that the representations U‖S (the restriction of U to S) and W↑G (the representation induced by W) are both orthogonal sums of finitely many irreducible subrepresentations, the number of which does not exceed p; (2) that the multiplicity of W in U‖S equals the multiplicity of U in W↑G if W and U are unitary representations and that these multiplicities are related in a slightly different manner for partially antiunitary representations. For the special case that S is an invariant subgroup, it is shown how the irreducible representations of G can be constructed if the irreducible representations of S and those of certain finite groups are known.