Multiplicity-free $U_q(sl_N)$ 6-j symbols: relations, asymptotics, symmetries

TL;DR

This paper provides a new expression for multiplicity-free quantum 6-j symbols of $U_q(sl_N)$ using q-hypergeometric series, explores their asymptotics, and uncovers new symmetries extending known $U_q(sl_2)$ properties.

Contribution

It introduces a universal approach to express MFS via $U_q(sl_2)$ 6-j symbols and derives their asymptotics and symmetries, generalizing known results.

Findings

Derived asymptotics of MFS in terms of classical algebra $U(sl_N)$

Expressed MFS using ${}_4 extPhi_3$ hypergeometric series

Identified new symmetry groups extending tetrahedral and Regge symmetries

Abstract

A closed form expression for multiplicity-free quantum 6-j symbols (MFS) was proposed in arXiv:1302.5143 for symmetric representations of $U_q(sl_N)$, 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 $U_q(sl_2)$ with a certain factor. It gives us a universal tool for the extension of various properties of the quantum 6-j symbols for $U_q(sl_2)$ to the MFS. We demonstrate this idea by deriving the asymptotics of the MFS in terms of associated tetrahedron for classical algebra $U(sl_N)$. 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.