Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesis

The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis various interesting properties of colored HOMFLY-PT polynomials will be proved. In addition, it allows one to discover new symmetries of the quantum 6-j symbols, about which almost nothing is known for $N>2$, with the exception of the tetrahedral symmetries, complex conjugation and transformation $q \longleftrightarrow q^{-1}$. In this paper we prove the eigenvalue hypothesis in $U_q(sl_2)$ case and show that it is equivalent to 6-j symbol symmetries (the Regge symmetry and two argument permutations). Then we apply the eigenvalue hypothesis to inclusive Racah matrices with 3 symmetric incoming representations of $U_q(sl_N)$ and an arbitrary outcoming one. It gives us 8 new additional symmetries that are not tetrahedral ones. Finally, we apply the eigenvalue hypothesis to exclusive Racah matrices with symmetric representations and obtain 4 tetrahedral symmetries.