# New symmetries for the $U_q(sl_N)$ 6-j symbols from the Eigenvalue conjecture

## TL;DR

This paper proves that the eigenvalue conjecture for $U_q(sl_2)$ Racah coefficients is equivalent to Regge symmetry, suggesting a pathway to generalize these symmetries to $U_q(sl_N)$ for arbitrary N.

## Contribution

It establishes the equivalence between the eigenvalue conjecture and Regge symmetry for $U_q(sl_2)$ and proposes a generalization to $U_q(sl_N)$.

## Key findings

- Eigenvalue conjecture is equivalent to Regge symmetry for $U_q(sl_2)$.
- Provides a basis for extending symmetries to $U_q(sl_N)$.
- Enhances understanding of symmetries in quantum group representations.

## Abstract

In the present paper we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of $U_q(sl_2)$. The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that \textbf{the eigenvalue conjecture is provided by the Regge symmetry} for $U_q(sl_2)$, when three representations coincide. This in perspective provides us a kind of generalization of the Regge symmetry to arbitrary $U_q(sl_N)$.