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

  • Gustafson R
  • Milne S
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We give a direct formulation of the invariant polynomialsμGq(n)(, Δi,;, xi,i + 1,) characterizing U(n) tensor operators 〈p, q, ..., q, 0, ..., 0〉 in terms of the symmetric functions Sλknown as Schur functions. To this end, we show after the change of variables Δi= γi- δiand xi, i + 1= δi- δi + 1thatμGq(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 thatμGq(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 polynomialsμmGq(n)(γ1,..., γn; δ1,..., δm). These new symmetries enable us to give an explicit formula for bothμmG1(n)(γ; δ) and1G2(n)(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas forμmGq(n)(γ; δ). © 1983.

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  • R. A. Gustafson

  • S. C. Milne

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