The Tetrahedral (or $6j$) Symbol

TL;DR

This paper generalizes the classical 6j symbol by attaching a scalar invariant to a tetrahedron with edges labeled by irreducible representations of SO(3), exploring its formulas, symmetries, and duality interpretations.

Contribution

It introduces a new scalar invariant for tetrahedra labeled by SO(3) representations, extending the 6j symbol theory with formulas, symmetry properties, and Langlands duality insights.

Findings

The invariant admits formulas involving hypergeometric integrals.

It exhibits symmetry under the Weyl group of Spin_{12}.

Connections to Langlands duality are established.

Abstract

We will attach a scalar invariant to a tetrahedron whose edges are labelled by irreducible representations of a ternary orthogonal group $\mathrm{SO}_3$ over a local field. This generalizes the $6j$ symbol whose theory was developed by Racah, Wigner, and Regge. We give several formulas for this invariant, including in terms of hypergeometric-type integrals and functions, and show that it admits a symmetry by the the $23040$-element Weyl group of $\mathrm{Spin}_{12}$. We then interpret these results in terms of relative Langlands duality, where the dual story comes from the action of $\mathrm{Spin}_{12}$ on a $16$-dimensional cone of spinors.