Polynomial realization of the U{sub q}{bold (}sl(3){bold )} Gel{close_quote}fand{endash}(Weyl){endash}Zetlin basis
- International Center for Theoretical Physics via Costiera 11, P.O. Box 586 34100 Trieste (Italy)
- Instituto Nazionale di Fisica Nucleare, Sezione di Genova (Italy)
An explicit realization of the scr(U){equivalent_to}U{sub q}{bold (}sl(3){bold )} Gel{close_quote}fand{endash}(Weyl){endash}Zetlin (GWZ) basis as polynomial functions in three variables (real or complex) is given. This realization is obtained in two complementary ways. First, a known correspondence is used between the abstract GWZ basis and explicit polynomials in the quantum subgroup scr(U){sup +} of the raising generators. Then an explicit construction is used of arbitrary lowest weight (holomorphic) representations of scr(U) in terms of three variables on which the generators of scr(U) are realized as q-difference operators. The application 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 {cflx I}{sup 2}, third component of isospin {cflx I}{sub z}, and hypercharge {cflx Y}, in the same realization as q-difference operators. The result is that the eigenvectors can be written in terms of q-hypergeometric polynomials in the three variables. Finally an explicit scalar product is constructed by adapting the Shapovalov form to this setting. The orthogonality of the GWZ polynomials with respect to this scalar product is proven using both realizations. This provides a polynomial construction for the orthonormal GWZ basis. The results here are for generic q, leaving the root of unity case for a following paper. It seems that the results are new also in the classical situation (q=1). {copyright} {ital 1997 American Institute of Physics.}
- OSTI ID:
- 530086
- Journal Information:
- Journal of Mathematical Physics, Vol. 38, Issue 7; Other Information: PBD: Jul 1997
- Country of Publication:
- United States
- Language:
- English
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