Recurrence relation for the 6j-symbol of su_q(2) as a symmetric eigenvalue problem

TL;DR

This paper reformulates the recurrence relation for the su_q(2) 6j-symbol as a symmetric eigenvalue problem, enabling efficient numerical computation and generalizing classical results using diagrammatic spin network formalism.

Contribution

It presents a novel symmetric eigenvalue problem formulation of the 6j-symbol recurrence relation, facilitating numerical evaluation and extending classical results to quantum groups.

Findings

Recurrence relation expressed as a tridiagonal symmetric eigenvalue problem

Implementation details for efficient numerical evaluation provided

Generalization of classical q=1 result using diagrammatic formalism

Abstract

A well known recurrence relation for the 6j-symbol of the quantum group su_q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking advantage of existing specialized numerical packages. For convenience, all formulas relevant for such an implementation are collected in the appendix. This realization is a byproduct of an alternative proof of the recurrence relation, which generalizes a classical (q=1) result of Schulten and Gordon and uses the diagrammatic spin network formalism of Temperley-Lieb recoupling theory to simplify intermediate calculations.