Multiple commutation relations of the quantum affine algebra $U_q({\widehat{\mathfrak{g}\mathfrak{l}}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis

We study a certain type of multiple commutation relations of the quantum affine algebra $U_q({\widehat{\mathfrak{g}\mathfrak{l}}}_N)$. 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 $Y_h({\mathfrak{g}\mathfrak{l}}_N)$.