We study a certain type of multiple commutation relations of the quantum affine algebra \(U_q(\widehat{\mathfrak {gl}}_N)\) . We show that all the coefficients in the multiple commutation relations between the L-operator elements are given in terms of the trigonometric weight functions for the vector representation, independent of the representation of the L-operator. For rank one case, our proof also gives a conceptual understanding why the coefficients can also be expressed using the Izergin–Korepin determinants. As a related result, by specializing expressions for the universal nested Bethe vector by Pakuliak–Ragoucy–Slavnov, we also find a construction of the Gelfand–Tsetlin basis for the vector representation using different L-operator elements from the constructions by Nazarov–Tarasov or Molev. We also present corresponding results for the Yangian \(Y_h(\mathfrak {gl}_N)\) .
