# Pentad and triangular structures behind the Racah matrices

## TL;DR

This paper uncovers a pentad structure behind Racah matrices related to twisted knots, revealing new algebraic decompositions and connections to Macdonald functions, which can enhance understanding in representation theory and knot invariants.

## Contribution

It introduces a novel pentad structure involving Racah matrices and triangular matrices, providing explicit formulas and connections to Macdonald functions for the first time.

## Key findings

- Discovery of a pentad structure $(ar T,ar S,S,{
m E},{
m B})$ behind Racah matrices.
- Explicit construction of the KNTZ evolution matrix ${
m B}$ using skew Schur and Macdonald functions.
- Potential implications for representation theory, knot invariants, and Macdonald-Kerov functions.

## Abstract

Somewhat unexpectedly, the study of the family of twisted knots revealed a hidden structure behind exclusive Racah matrices $\bar S$, which control non-associativity of the representation product in a peculiar channel $R\otimes \bar R \otimes R \longrightarrow R$. These $\bar S$ are simultaneously symmetric and orthogonal, and therefore admit two decompositions: as quadratic forms, $\bar S \sim {\cal E}^{tr}{\cal E}$, and as operators: $\bar T\bar S\bar T = S T^{-1} S^{-1}$. Here $\bar T$ and $T$ consist of the eigenvalues of the quantum ${\cal R}$-matrices in channels $R\otimes \bar R$ and $R\otimes R$ respectively, $S$ is the second exclusive Racah matrix for $\bar R\otimes R\otimes R \longrightarrow R$ (still orthogonal, but no longer symmetric), and ${\cal E}$ is a {\it triangular} matrix. It can be further used to construct the KNTZ evolution matrix ${\cal B}={\cal E}\bar T^2{\cal E}^{-1}$, which is also triangular and explicitly expressible through the skew Schur and Macdonald functions -- what makes Racah matrices calculable. Moreover, ${\cal B}$ is somewhat similar to Ruijsenaars Hamiltonian, which is used to define Macdonald functions, and gets triangular in the Schur basis. Discovery of this pentad structure $(\bar T,\bar S,S,{\cal E},{\cal B})$, associated with the universal ${\cal R}$-matrix, can lead to further insights about representation theory, knot invariants and Macdonald-Kerov functions.