Calculation of $6j$-symbols for the Lie algebra $\mathfrak{gl}_n$

TL;DR

This paper provides an explicit formula for $6j$-symbols of the Lie algebra $rak{gl}_n$, expressing them via generalized hypergeometric functions, advancing the understanding of tensor product decompositions.

Contribution

It introduces a new explicit formula for $rak{gl}_n$ $6j$-symbols based on multiplicity space descriptions and hypergeometric functions.

Findings

Explicit $6j$-symbol formula for $rak{gl}_n$

Expression of $6j$-symbols through hypergeometric functions

Enhanced understanding of tensor product decompositions

Abstract

An explicit description of the multiplicity space that describes occurrences of irreducible representations in a splitting of a tensor product of two irreducible representations of $\mathfrak{gl}_n$ is given. Using this description an explicit formula for an arbitrary $6j$-symbol for the algebra $\mathfrak{gl}_n$ is derived. The $6j$-symbol is expressed through a value of a generalized hypergeometric function.