Irregular Uq(sl(3)) representations at roots of unity via Gel-fand-(Weyl)-Zetlin basis

We give explicitly the Gel'fand-(Weyl)Zetlin (GWZ) description of the Uq(sl(3)) irregular irreps at roots of unity. Those are irreps fixed by the same parameters as the UIRs of SU(3) yet having dimensions smaller than their classical counterparts, the reason being that to obtain an irregular irrep we have to make factorization of an additional submodule. This description is made geometrically transparent by an arrangement of the standard SU(3) GWZ basis in a hexagonal pyramid (valid for any q). The pyramid has as a base the standard hexagon which gives the weight space of the UIRs of SU(3) in the plane of third component of isospin Iz and hypercharge Y, while third dimension of our pyramid is related to the isospin I. Algebraically this arrangement is related to a 1-to-1 correspondence between the abstract GWZ states and monomials in the algebra of raising generators Uq(G+), however, those monomials being not in the standard Poincare-Birkhoff-Witt basis of Uq(G+). The additional factorization corresponds to taking away an upper part of the pyramid, itself being a hexagonal pyramid representing another SU(3) irrep of smaller dimension, which for roots of unity becomes the submodule to be factored out. The technical tool 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 our pyramid. (author). 27 refs