Abstract
We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrixB, 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.
Highlights
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]
The KNTZ claim (15) is that the switch from a diagonal evolution matrix T2 to triangular B, though looks like a complication, reveals the hidden structure of the differential expansion for twist knots and somehow trades the sophisticated Racah matrix Sfor a much simpler and universal B. The reason for this can be that the actual evolution matrix was not the simple diagonal T2, but rather a sophisticated symmetric ST2S, and the switch to triangular B is a simplification. In this approach the crucial role is played by the switching matrix U, and the central phenomenon is a drastic simplicity of the first line in a peculiar matrix U T2U −1B−1: for rectangular representations its entries are just products of the differentials, and better understanding of the phenomenon can help to explain the linear combinations of those, which emerge in the non-rectangular case
Suggested in [12] was a complicated version, based on the possibility to re-express skew Schur functions at the zero-locus through shifted Schur functions [29]– so that the β-deformation involves shifted Macdonald polynomials [30]. This works, but there is a much simpler option: transposed skew Schurs in (16) can be substituted by those of negative times – and they can be substituted by skew Macdonalds
Summary
The reason for this can be that the actual evolution matrix was not the simple diagonal T2, but rather a sophisticated symmetric ST2S, and the switch to triangular B is a simplification. In this approach the crucial role is played by the switching matrix U , and the central phenomenon is a drastic simplicity of the first line in a peculiar matrix U T2U −1B−1: for rectangular representations its entries are just products of the differentials, and better understanding of the phenomenon can help to explain the linear combinations of those, which emerge in the non-rectangular case.
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