Unitary group subjoinings

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Abstract
The subjoining of one compact Lie group H to another such group G is discussed with particular reference to the cases for which G=U(N) and H=U(n). It is shown that maximal subjoinings of these unitary groups are specified by means of the monomial symmetric functions. Subjoinings, which are defined in terms of mappings between weight spaces, are studied through the properties of characters of the irreducible representations. The branching rules corresponding to subjoinings are found to involve plethysms. Methods of evaluating the appropriate plethysms are illustrated, some of which make use of subjoining chains whilst others exploit the Weyl symmetry groups of G and H to obtain results more directly. The fact that maximal embeddings are special cases of non-maximal subjoinings is demonstrated and discussed.

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