# On exclusive Racah matrices $\bar S$ for rectangular representations

## TL;DR

This paper presents a universal formula for the triangular evolution matrix ${\

## Contribution

It introduces a universal expression for the matrix ${\cal B}$ applicable to all rectangular representations and relates it to Racah matrices and differential expansion coefficients.

## Key findings

- Universal formula for ${\cal B}$ in rectangular representations
- Explicit Racah matrix ${\bar S}$ derived from matrix evolution
- Relation between matrix evolution and Racah matrix rotation U

## Abstract

We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrix ${\cal B}$, not just its eigenvalues $\Lambda$, and provide a universal formula for ${\cal B}$, 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 $R$.