Representations of U(1,q) and Constructive Quaternion Tensor Products

The representation theory of the group U(1,q) is discussed in detail because of its possible application in a quaternion version of the Salam-Weinberg theory. As a consequence, from purely group theoretical arguments we demonstrate that the eigenvalues must be right-eigenvalues and that the only consistent scalar products are the complex ones. We also define an explicit quaternion tensor product which leads to a set of additional group representations for integer ``spin''.