Schur functions and the invariant polynomials characterizing U(n) tensor operators

Abstract

We give a direct formulation of the invariant polynomials μGq(n)(, Δi,;, xi,i + 1,) characterizing U(n) tensor operators 〈p, q, ..., q, 0, ..., 0〉 in terms of the symmetric functions Sλ known as Schur functions. To this end, we show after the change of variables Δi = γi - δi and xi, i + 1 = δi - δi + 1 thatμGq(n)(,Δi;, xi, i + 1,) becomes an integral linear combination of products of Schur functions Sα(, γi,) · Sβ(, δi,) in the variables {γ1,..., γn} and {δ1,..., δn}, respectively. That is, we give a direct proof that μGq(n)(,Δi,;, xi, i + 1,) is a bisymmetric polynomial with integer coefficients in the variables {γ1,..., γn} and {δ1,..., δn}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials μmGq(n)(γ1,..., γn; δ1,..., δm). These new symmetries enable us to give an explicit formula for both μmG1(n)(γ; δ) and 1G2(n)(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for μmGq(n)(γ; δ). © 1983.