Bases for the Irreducible Representations of the Unitary Groups and Some Applications

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Abstract
In this paper we show that sets of polynomials in the components of (2j + 1)‐dimensional vectors, solutions of certain invariant partial differential equations, form bases for all the irreducible representations of the unitary group U2j+1. These polynomials will play, for the group U2j+1, the same role that the solid spherical harmonics (themselves polynomials in the components of a three‐dimensional vector) play for the rotation group R3. With the help of these polynomials we define and determine the reduced Wigner coefficients for the unitary groups, which we then use to derive the Wigner coefficients of U2j+1 by a factorization procedure. An ambiguity remains in the explicit expression for the Wigner coefficients as the Kronecker product of two irreducible representations of U2j+1 is not, in general, multiplicity‐free. We show how to eliminate this ambiguity with the help of operators that serve to characterize completely the rows of representations of unitary groups for a particular chain of subgroups. The procedure developed to determine the polynomial bases of U2j+1 seems, in principle, generalizable to arbitrary semisimple compact Lie groups.