POLYNOMIAL REALIZATION OF THE U-Q(SL(3)) GELFAND-(WEYL)-ZETLIN BASIS

An explicit realization of the U=U-q(sl(3)) Gel'fand-(Weyl)-Zetlin (GW Z) basis as polynomial functions in three variables (real or complex) is given. This realization is obtained in two complementary ways. Firs t, a known correspondence is used between the abstract GWZ basis and e xplicit polynomials in the quantum subgroup U+ of the raising generato rs. Then an explicit construction is used of arbitrary lowest weight ( holomorphic) representations of U in terms of three variables on which the generators of U are realized as q-difference operators. The appli cation of the GWZ corresponding polynomials in this realization of the lowest weight vector (the function 1) produces the first realization of this GWZ basis, Another realization of the GWZ polynomial basis is found by the explicit diagonalization of the operators of isospin (I) over cap(2), third component of isospin (I) over cap(z), and hyperchar ge (Y) over cap, in the same realization as q-difference operators. Th e result is that the eigenvectors can be written in terms of q-hyperge ometric polynomials in the three variables, Finally an explicit scalar product is constructed by adapting the Shapovalov form to this settin g, The orthogonality of the GWZ polynomials with respect to this scala r product is proven using both realizations, This provides a polynomia l construction for the orthonormal GWZ basis. The results hen are for generic q, leaving the root of unity case for a following paper. II se ems that the results are new also in the classical situation (q = 1). (C) 1997 American Institute of Physics.