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

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.

7 pages

References in corpus (2)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations

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• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Eigenvalue hypothesis for multi-strand braids
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Distinguishing Mutant Knots
• Checks of integrality properties in topological strings
• New symmetries for the $U_q(s{l}_N)$ 6-j symbols from the Eigenvalue conjecture
• Extension of KNTZ trick to non-rectangular representations
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• On $6j$-symbols for symmetric representations of $U_q({\mathfrak{su}}_N)$
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• On the block structure of the quantum R-matrix in the three-strand braids
• Difference of mutant knot invariants and their differential expansion
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• On knots, complements, and 6j-symbols
• Tug-the-hook symmetry for quantum 6j-symbols
• Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism