Schur functions and the invariant polynomials characterizing U(n) tensor operators
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We give a direct formulation of the invariant polynomials Gq(n)(, i,;, xi,i + 1,) characterizing U(n) tensor operators p, q, ..., q, 0, ..., 0 in terms of the symmetric functions S known as Schur functions. To this end, we show after the change of variables i = i - i and xi, i + 1 = i - i + 1 thatGq(n)(,i;, xi, i + 1,) becomes an integral linear combination of products of Schur functions S(, i,) S(, i,) in the variables {1,..., n} and {1,..., n}, respectively. That is, we give a direct proof that Gq(n)(,i,;, xi, i + 1,) is a bisymmetric polynomial with integer coefficients in the variables {1,..., n} and {1,..., n}. 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 mGq(n)(1,..., n; 1,..., m). These new symmetries enable us to give an explicit formula for both mG1(n)(; ) and 1G2(n)(; ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for mGq(n)(; ). 1983.