On exclusive Racah matrices 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 B, not just its eigenvalues Λ, and provide a universal formula for B, applicable to arbitrary rectangular representation R=[rs]. 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 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.