An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

TL;DR

This paper presents an elementary construction of elliptic 6j-symbols, unifying various special cases and providing new algebraic insights through Sklyanin algebra representations.

Contribution

It introduces a simple approach to elliptic 6j-symbols that clarifies their properties and links them to Sklyanin algebra representations, expanding understanding of their structure.

Findings

Elementary construction of elliptic 6j-symbols

Derivation of key properties of elliptic 6j-symbols

New algebraic interpretation via Sklyanin algebra

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.