Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries

Abstract

A closed form expression for multiplicity-free quantum 6-j symbols (MFS) was proposed in arXiv:1302.5143 for symmetric representations of $U_q(s{l}_N)$, which are the simplest class of multiplicity-free representations. In this paper we rewrite this expression in terms of q-hypergeometric series ${}_4{\Phi }_3$. We claim that it is possible to express any MFS through the 6-j symbol for $U_q(s{l}_2)$ with a certain factor. It gives us a universal tool for the extension of various properties of the quantum 6-j symbols for $U_q(s{l}_2)$ to the MFS. We demonstrate this idea by deriving the asymptotics of the MFS in terms of associated tetrahedron for classical algebra $U(s{l}_N)$. Next we study MFS symmetries using known hypergeometric identities such as argument permutations and Sears' transformation. We describe symmetry groups of MFS. As a result we get new symmetries, which are a generalization of the tetrahedral symmetries and the Regge symmetries for N = 2.

arXiv comment25 pages

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