On the block structure of the quantum R-matrix in the three-strand braids

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 SUq(N) associated with each strand, one needs two matrices: ℛ1 and ℛ2. They are related by the Racah matrices ℛ2 = 𝒰ℛ1𝒰†. Since we can always choose the basis so that ℛ1 is diagonal, the problem is reduced to evaluation of ℛ2-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that ℛ2-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 ℛ1 matrix. In this case in order to get a block-diagonal matrix, one should rotate the ℛ2 defined by the Racah matrix in the accidental sector by the angle exactly ±π 4.