On exclusive Racah matrices $\bar{S}$ for rectangular representations

Abstract

We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrix $\mathcal{B}$, not just its eigenvalues $\Lambda$, and provide a universal formula for $\mathcal{B}$, applicable to arbitrary rectangular representation $R=[r^s]$. This expression is in terms of skew characters and it remains literally the same for the 4-graded rectangularly-colored hyperpolynomials, if characters are substituted by Macdonald polynomials. Due to additional factorization property of the differential-expansion coefficients for the double-braid knots, explicit knowledge of twist-family evolution leads to a nearly explicit answer for Racah matrix $\bar{S}$ in arbitrary rectangular representation $R$. We also relate matrix evolution to existence of a peculiar rotation $U$ of Racah matrix, which diagonalizes the $Z$-factors in the differential expansion -- what can be a key to further generalization to non-rectangular representations $R$.

12 pages

References in corpus (11)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• On AGT relation in the case of U(3)
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Refined Chern-Simons Theory and Topological String
• On colored HOMFLY polynomials for twist knots
• Refined Chern-Simons Theory and Knot Homology
• On rectangular HOMFLY for twist knots
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Extension of KNTZ trick to non-rectangular representations
• Kerov functions for composite representations and Macdonald ideal

Cited by in corpus (1)

• Extension of KNTZ trick to non-rectangular representations