Multiple Commutation Relations of the Quantum Affine Algebra $$U_q(\widehat{\mathfrak {gl}}_N)$$, Nested Bethe Vector, and the Gelfand–Tsetlin Basis

Abstract We study a certain type of multiple commutation relations of the quantum affine algebra $$U_q(\widehat{\mathfrak {gl}}_N)$$ U q ( 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 $$Y_h(\mathfrak {gl}_N)$$ Y h ( gl N ) .