关于矩形表示的排他Racah矩阵$\bar S$
On exclusive Racah matrices $\bar S$ for rectangular representations
摘要：
我们阐述了最近的观察，扭转结的演变简化三角形的演变矩阵${\cal B}$，而不仅仅是其特征值$Λ$，并提供了一个通用的公式${\cal B}$，适用于任意矩形表示$R=[r^s]$。该表达式是斜特征的表示，如果用Macdonald多项式代替特征，则对于4阶矩形着色超多项式，它在字面上保持相同。由于双辫纽结的微分展开系数具有额外的因式分解性质，明确的扭族演化知识导致了任意矩形表示$R$中Racah矩阵$\bar S$的近似明确的答案.我们还涉及矩阵演化的存在一个特殊的旋转$U$的拉卡矩阵，对角化的$Z$ -因素的微分展开-什么可以是一个关键，进一步推广到非矩形表示$R$。
英文摘要：
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 $Λ$, 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 $R$.
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