Quantum Racah matrices and 3-strand braids in irreps R with |R|=4

TL;DR

This paper computes inclusive Racah matrices for the first non-symmetric rectangular representation in quantum groups, enabling detailed calculations of colored HOMFLY polynomials for complex knots.

Contribution

It provides the first complete description of Racah matrices for |R|=4 in quantum groups, including new methods for degenerate cases and extraction of exclusive matrices.

Findings

Complete Racah matrices for R=[2,2] with sizes 2, 3, 4, and 6x6 cases.

Method for handling degenerate eigenvalues using the highest weight approach.

Facilitates calculation of colored HOMFLY polynomials for 3-strand and arborescent knots.

Abstract

We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R=[2,2] for quantum groups U_q(sl_N). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6x6 matrices, one is also described in this way, but the other one corresponds to the case of degenerate eigenvalues and is evaluated by the highest weight method. Together with the much harder calculation for R=[3,1] in arXiv:1605.02313 and with the new method to extract exclusive matrices S and \bar S from the inclusive ones, this completes the story of Racah matrices for |R|\leq 4 and allows one to calculate and investigate the corresponding colored HOMFLY polynomials for arbitrary 3-strand and arborescent knots.