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Interplay between symmetries of quantum 6j-symbols and the eigenvalue hypothesis
Letters in Mathematical Physics ( IF 1.3 ) Pub Date : 2021-04-18 , DOI: 10.1007/s11005-021-01386-1
Victor Alekseev , Andrey Morozov , Alexey Sleptsov

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.



中文翻译:

量子6j符号的对称性与特征值假设之间的相互作用

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。

更新日期:2021-04-18
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