# Mathematical proof of Regge symmetry

In the representation theory of the group $SU_2$ a big role is played by so-called $6j-$symbols. Let me sketch its definition (some other interpretations could be found here).

Denote a representation of highest weight $a$ by $V_a.$ The space of invariants $$Inv(V_a \otimes V_b \otimes V_c \otimes V_d)$$ has two natural bases coming from pairings $$Inv(V_a \otimes V_b \otimes V_e) \otimes Inv(V_e^* \otimes V_c \otimes V_d ) \longrightarrow Inv(V_a \otimes V_b \otimes V_c \otimes V_d)$$ and $$Inv(V_a \otimes V_c \otimes V_f) \otimes Inv(V_f^* \otimes V_b \otimes V_d ) \longrightarrow Inv(V_a \otimes V_b \otimes V_c \otimes V_d)$$ for various $e$ and $f$. A $6j-$symbol $$\begin{bmatrix} a & b & c\\ d & e & f\\ \end{bmatrix}$$ is the corresponding coefficient of an operator on the space $$Inv(V_a \otimes V_b \otimes V_c \otimes V_d),$$ sending the first basis to the second.

These numbers poses a surprising number of symmetries. The most surprising one is the famous Regge symmetry: $$\begin{bmatrix} a & b & c\\ d & e & f\\ \end{bmatrix}= \begin{bmatrix} a & s-b & s-c\\ d & s-e & s-f\\ \end{bmatrix},$$ where $$s=\frac{b+c+e+f}{2}.$$

Question: How to prove this statement? In physical books it is usually derived as a corollary of a direct computation. Also, any mathematical interpretation of these symmetries will be of great interest to me.