Unitary Groups: Representations and Decompositions

Abstract

An elementary account is given of the representation theory for unitary groups. We review the basic definitions and the construction of irreducible representations using tensor methods, and indicate the connection to the infinitesimal approach. Special attention has been given to the detailed procedure to obtain Clebsch-Gordan series and to the problem of finding the ($S{U}_m, S{U}_n$) content of an irreducible representation of $S{U}_{\mathrm{mn}}$ or $S{U}_{m+n}$. An appendix summarizes the properties of the Young operators used in constructing the tensor representations; this provides the link with the representation theory of the symmetric groups. We include a tabulation of various decompositions which appear in the text and of Weyl's dimension formula for tensor representation.