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The Tetrahedral (or $6j$) Symbol

Akshay Venkatesh, X. Griffin Wang

TL;DR

The paper generalizes the classical $6j$ symbol to a tetrahedral invariant attached to tetrahedral edge-labelings by representations of SO$_3$ over a local field, uncovering a rich structure governed by the Weyl group of Spin$_{12}$ and Langlands duality. It develops two complementary viewpoints: (i) a concrete analytic framework using vertex and edge integrals and hypergeometric-type formulas that express the tetrahedral symbol in terms of gamma-factors, trace formulas on spinor cones, and moduli of points on projective lines; and (ii) a Langlands-dual interpretation linking the invariant to the cone of pure spinors in Spin$_{12}$ and to geometric representation theory. The main results establish a $W(\mathsf{D}_6)$-symmetry for principal-series inputs and provide an explicit unramified evaluation via the spinor cone, thereby completing a conceptual bridge between harmonic analysis on local fields and the geometry of Spin$_{12}$. The work also develops extensive hypergeometric formulations, orthogonality properties, and sign-resolution mechanisms, with broad implications for relative Langlands duality and potential future connections to automorphic forms and geometric representation theory.

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.

The Tetrahedral (or $6j$) Symbol

TL;DR

The paper generalizes the classical symbol to a tetrahedral invariant attached to tetrahedral edge-labelings by representations of SO over a local field, uncovering a rich structure governed by the Weyl group of Spin and Langlands duality. It develops two complementary viewpoints: (i) a concrete analytic framework using vertex and edge integrals and hypergeometric-type formulas that express the tetrahedral symbol in terms of gamma-factors, trace formulas on spinor cones, and moduli of points on projective lines; and (ii) a Langlands-dual interpretation linking the invariant to the cone of pure spinors in Spin and to geometric representation theory. The main results establish a -symmetry for principal-series inputs and provide an explicit unramified evaluation via the spinor cone, thereby completing a conceptual bridge between harmonic analysis on local fields and the geometry of Spin. The work also develops extensive hypergeometric formulations, orthogonality properties, and sign-resolution mechanisms, with broad implications for relative Langlands duality and potential future connections to automorphic forms and geometric representation theory.

Abstract

We will attach a scalar invariant to a tetrahedron whose edges are labelled by irreducible representations of a ternary orthogonal group over a local field. This generalizes the 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 -element Weyl group of . We then interpret these results in terms of relative Langlands duality, where the dual story comes from the action of on a -dimensional cone of spinors.
Paper Structure (99 sections, 36 theorems, 377 equations, 1 figure, 1 table)

This paper contains 99 sections, 36 theorems, 377 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The classical definition.
  3. What we do in this paper, and why.
  4. What?
  5. Why?
  6. Connection to existing work
  7. A summary of the paper
  8. Acknowledgements
  9. Review of harmonic analysis on a local field
  10. Characters, Haar measures, and absolute values
  11. $\gamma$, $L$, and $\epsilon$-factors
  12. The $\gamma$-factor as the Fourier transform of a quasi-character
  13. Evaluation of the $\gamma$-factor
  14. Other equalities for the $\gamma$-factor
  15. Multisets of characters
  16. ...and 84 more sections

Key Result

Lemma 3.4.2

[lemma]PrasadLemma There exists a real structure $\pi_e^{\mathbb{R}}$ on which $\IPair{-}{-}$ defines a real inner product.

Figures (1)

  • Figure 1: A tetrahedron with labeled edges

Theorems & Definitions (77)

  • Remark 1.1.1
  • Definition 2.1.1
  • Remark 2.2.2
  • Lemma 3.4.2
  • proof
  • Proposition 3.5.3
  • Remark 3.5.4
  • Remark 3.5.5
  • Lemma 4.3.1
  • Lemma 4.3.2
  • ...and 67 more