IRREGULAR U-Q(SL(3)) REPRESENTATIONS AT ROOTS OF UNITY VIA GELFAND-(WEYL)-ZETLIN BASIS

The Gel'fand-(Weyl)-Zetlin (GWZ) description of the U-q(sl(3)) irregul ar irreps at roots of unity is explicitly given. Those are irreps fixe d by the same parameters as the unitary irreducible representations (U IRs) of SU(3) yet having dimensions smaller than their classical count erparts, the reason being that to obtain an irregular irrep one has to make factorization of an additional submodule. This description is ma de geometrically transparent by an arrangement of the standard SU(3) G WZ basis in a hexagonal pyramid, which is valid for any q and seems ne w even for q=1. The pyramid has as a base the standard hexagon which g ives the weight space of the UIRs of SU(3) in the plane of third compo nent of isospin I-z and hypercharge Y, while third dimension of this p yramid is related to the isospin I. Algebraically this arrangement is related to a one-to-one correspondence between the abstract GWZ states and monomials in the algebra of raising generators U-q(G(+)); however , those monomials are not in the standard Poincare-Birkhoffs-Witt (PBW ) basis of U-q(G(+)). The additional factorization corresponds to taki ng away an upper part of the pyramid, itself being a hexagonal pyramid representing another SU(3) irrep of smaller dimension, which for root s of unity becomes the submodule to be factored out. The technical too l in this factorization is the explicit coincidence of two polynomials : one giving the singular vectors of the Verma modules, the other used in the algebraic description of the pyramid. (C) 1997 American Instit ute of Physics.