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

Abstract

Quantum $\mathcal{R}$-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: ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$. They are related by the Racah matrices ${\mathcal{R}}_2=\mathcal{U}{\mathcal{R}}_1{\mathcal{U}}^{†}$. Since we can always choose the basis so that ${\mathcal{R}}_1$ is diagonal, the problem is reduced to evaluation of ${\mathcal{R}}_2$-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that ${\mathcal{R}}_2$-matrices could be transformed into a block-diagonal ones. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of ${\mathcal{R}}_1$-matrix. The angle of the rotation in the sectors corresponding to accidentally coinciding eigenvalues from the basis defined by the Racah matrix to the basis in which ${\mathcal{R}}_2$ is block-diagonal is $\pm \frac{\pi }{4}$.

21 pages