New Symmetries for the U q ( sl N ) 6- j Symbols from the Eigenvalue Conjecture1

In the present paper, we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of Uq(slN) 6-j The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that the eigenvalue conjecture is provided by the Regge symmetry for Uq(slN) 6-j, when three representations coincide. This in perspective provides us a kind of generalization of the Regge symmetry to arbitrary Uq(slN) 6-j.