Interplay between symmetries of quantum 6j-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 $$\mathcal {R}$$ -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 6j-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 6j-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.