AN UMBRAL CALCULUS FOR POLYNOMIALS CHARACTERIZING-U(N) TENSOR-OPERATORS
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After the change of variables i = i - i and xi,i + 1 = i - i + 1 we show that the invariant polynomials G(n)q(, i, ;, xi,i+1,) characterizing U(n) tensor operators p, q,..., q, 0,..., 0 become an integral linear combination of Schur functions S( - ) in the symbol - , where - denotes the difference of the two sets of variables {1,..., n} and {1,..., n}. We obtain a similar result for the yet more general bisymmetric polynomials mG(n)q(1,..., n; 1,..., m). Making use of properties of skew Schur functions S and S( - ) we put together an umbral calculus for mG(n)q(; ). That is, working entirely with polynomials, we uniquely determine mG(n)q(; ) from mG(n)q - 1(; ) and combinatorial rules involving Ferrers diagrams (i.e., partitions), provided that n ( + 1)q. (This restriction does not interfere with writing the general case of mG(n)q(; ) as a linear combination of S( - ).) As an application we deduce "conjugation" symmetry for nG(n)q(; ) from "transposition" symmetry by showing that these two symmetries are equivalent. 1984.