Abstract. Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson's biorthogonal 10 W 9 functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6j-symbols in terms of Sklyanin algebra representations.
IntroductionThe classical 6j-symbols were introduced by Racah and Wigner in the early 1940's [Rac, Wi]. Though they appeared in the context of quantum mechanics, they are natural objects in the representation theory of SL(2) that can be introduced from purely mathematical considerations. Wilson [W1] realized that 6j-symbols are orthogonal polynomials, and that they generalize many classical systems such as Krawtchouk and Jacobi polynomials. This led Askey and Wilson to introduce the more general q-Racah polynomials [AW1].The q-Racah polynomials belong to the class of basic (or q-) hypergeometric series [GR1]. Since the 1980's, there has been a considerable increase of interest in this classical subject. One reason for this is relations to solvable models in statistical mechanics, and to the related algebraic structures known as quantum groups.Kirillov and Reshetikhin [KR] found that q-Racah polynomials appear as 6j-symbols of the SL(2) quantum group, or quantum 6j-symbols. We mention that in the introduction to the standard reference [CP], three major applications of quantum groups to other fields of mathematics are highlighted. For at least two of these, namely, invariants of links and three-manifolds [Tu], and the relation to affine Lie algebras and conformal field theory [EFK], quantum 6j-symbols play a decisive role.The q-Racah polynomials form, together with the closely related Askey-Wilson polynomials, the top level of the Askey Scheme of (q-)hypergeometric orthogonal polynomials [KS]. One reason for viewing this scheme as complete is Leonard's 1991 Mathematics Subject Classification. 33D45, 33D80, 82B23.