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The eigenvalue hypothesis claims that any quantum Racah matrix forfinite-dimensional representations of $U_q(sl_N)$ is uniquely determined byeigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesisturns out to be true, then it will significantly simplify the computation ofRacah matrices. Also due to this hypothesis various interesting properties ofcolored HOMFLY-PT polynomials will be proved. In addition, it allows one todiscover new symmetries of the quantum 6-j symbols, about which almost nothingis known for $N>2$, with the exception of the tetrahedral symmetries, complexconjugation and transformation $q \longleftrightarrow q^{-1}$.  In this paper we prove the eigenvalue hypothesis in $U_q(sl_2)$ case and showthat it is equivalent to 6-j symbol symmetries (the Regge symmetry and twoargument permutations). Then we apply the eigenvalue hypothesis to inclusiveRacah matrices with 3 symmetric incoming representations of $U_q(sl_N)$ and anarbitrary outcoming one. It gives us 8 new additional symmetries that are nottetrahedral ones. Finally, we apply the eigenvalue hypothesis to exclusiveRacah matrices with symmetric representations and obtain 4 tetrahedralsymmetries.

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