Regge symmetry and partition of Wigner 3-j or super 3-jS symbols: unknown properties

Abstract

For each generic $(3\mbox{-}j)$ the column parities, $2(j\pm m)$, define $3$ intrinsic parities: $\alpha ,\beta ,\gamma$. In algebra $so(3)$ only $(3\mbox{-}j)\_α$ exists whereas super-algebra $osp(1|2)$ admits $3$ kinds of super-symbols $(3\mbox{-}j)^{S}\_α, (3\mbox{-}j)^{S}\_β, (3\mbox{-}j)^{S}\_γ$. Instead of 4 for $\{6\mbox{-}j\}$ symbols, Regge symmetry this time produces 5 partitions $\mathtt{S}\_\wr (0),\mathtt{S}\_\wr (1),\mathtt{S}\_\wr (2),\mathtt{S}\_\wr (4),\mathtt{S}\_\wr (5)$, with $\mathtt{S}\_\wr (3)=\varnothing$. Valid for $(3\mbox{-}j)\_α, (3\mbox{-}j)^{S}\_{α,γ}$ they reduce to 2 for $(3\mbox{-}j)^{S}\_β$ with $\mathtt{S}\_\wr (0),\mathtt{S}\_\wr (1)$. Unexpectedly a symbol $(3\mbox{-}j)^{S}\_β$ and its 'Regge-transformed' may be opposite in sign. In terms of integer parts and super-triangle ${△}^S$ a formula similar to that of a $(3\mbox{-}j)$ is obtained for the $(3\mbox{-}j)^{S}$. Some forbidden $(3\mbox{-}j)^{S}\_β$ require an analytic prolongation, consistent with Regge $\beta$-partitions.

arXiv comment12 pages