Quantum $ℛ$ -matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation $T$ of $S U_{q} (N)$ associated with each strand, one needs two matrices: $ℛ_{1}$ and $ℛ_{2}$ . They are related by the Racah matrices $ℛ_{2} = 𝒰 ℛ_{1} 𝒰^{†}$ . Since we can always choose the basis so that $ℛ_{1}$ is diagonal, the problem is reduced to evaluation of $ℛ_{2}$ -matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that $ℛ_{2}$ -matrices could be transformed into a block-diagonal ones by the rotation in the sectors of coinciding eigenvalues. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of $ℛ_{1}$ matrix. In this case in order to get a block-diagonal matrix, one should rotate the $ℛ_{2}$ defined by the Racah matrix in the accidental sector by the angle exactly $\pm \frac{π}{4}$ .

Keywords:

PACS: 02.10.Kn, 11.15.Yc

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