The Tetrahedral (or $6j$) Symbol
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911450757332992 |
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| author | Venkatesh, Akshay Wang, X. Griffin |
| author_facet | Venkatesh, Akshay Wang, X. Griffin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14908 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Tetrahedral (or $6j$) Symbol Venkatesh, Akshay Wang, X. Griffin Number Theory Representation Theory 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. |
| title | The Tetrahedral (or $6j$) Symbol |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2602.14908 |