Bases for the Irreducible Representations of the Unitary Groups and Some Applications
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 U 2j+1. These polynomials will play, for the group U 2j+1, the same role that the solid spherical harmonics (themselves polynomials in the components of a three-dimensional vector) play for the rotation group R 3. 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 U 2j+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 U 2j+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 U 2j+1 seems, in principle, generalizable to arbitrary semisimple compact Lie groups.
- Research Organization:
- Universidad de Mexico, Mexico City
- Sponsoring Organization:
- USDOE
- NSA Number:
- NSA-17-036841
- OSTI ID:
- 4632075
- Journal Information:
- Journal of Mathematical Physics (New York) (U.S.), Vol. Vol: 4; Other Information: Orig. Receipt Date: 31-DEC-63
- Country of Publication:
- Country unknown/Code not available
- Language:
- English
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