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 $(3\mbox{-}j)\_{\alpha}$ exists whereas super-algebra $osp(1|2)$ admits $3$ kinds of super-symbols $(3\mbox{-}j)^{S}\_{\alpha}, (3\mbox{-}j)^{S}\_{\beta}, (3\mbox{-}j)^{S}\_{\gamma}$. Instead of 4 for $\{6\mbox{-}j\}$ symbols, Regge symmetry this time produces 5 partitions ${\tt S}\_{\boldsymbol{\wr}}(0), {\tt S}\_{\boldsymbol{\wr}}(1), {\tt S}\_{\boldsymbol{\wr}}(2), {\tt S}\_{\boldsymbol{\wr}}(4), {\tt S}\_{\boldsymbol{\wr}}(5)$, with ${\tt S}\_{\boldsymbol{\wr}}(3) = \emptyset$. Valid for $(3\mbox{-}j)\_{\alpha}, (3\mbox{-}j)^{S}\_{\alpha,\gamma}$ they reduce to 2 for $(3\mbox{-}j)^{S}\_{\beta}$ with ${\tt S}\_{\boldsymbol{\wr}}(0), {\tt S}\_{\boldsymbol{\wr}}(1)$. Unexpectedly a symbol $(3\mbox{-}j)^{S}\_{\beta}$ and its 'Regge-transformed' may be opposite in sign. In terms of integer parts and super-triangle $\triangle^{S}$ a formula similar to that of a $(3\mbox{-}j)$ is obtained for the $(3\mbox{-}j)^{S}$. Some forbidden $(3\mbox{-}j)^{S}\_{\beta}$ require an analytic prolongation, consistent with Regge $\beta$-partitions.