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.

本征值假设声称，用于\（U_q（sl_N）\）的有限维表示的任何量子Racah矩阵都是由相应的量子\（\ mathcal {R} \）矩阵的特征值唯一确定的。如果该假设成立，那么它将大大简化Racah矩阵的计算。同样，由于这个假设，将证明彩色HOMFLY-PT多项式的各种有趣性质。此外，它还可以发现量子6j符号的新对称性，关于\（N> 2 \）几乎一无所知，除了四面体对称性，复杂的共轭和变换\（q \ longleftrightarrow q ^ {-1} \）。在本文中，我们证明了特征值假设\（U_q（sl_2）\）并显示它等效于6j-符号对称（Regge对称和两个参数置换）。然后，我们将特征值假设应用于包含\（U_q（sl_N）\）的3个对称传入表示和任意结果的包容Racah矩阵。它给了我们8个非四面体的新对称性。最后，我们将特征值假设应用于具有对称表示的排他性Racah矩阵，并获得4个四面体对称性。