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 $U_q({\widehat{\mathfrak{gl}}}_N)$. 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 $Y_h({\mathfrak{gl}}_N)$.

44 pages, 23 figures. Ann. Henri Poincaré (2026) Added revisions suggested by the reviewers. - Fixed typos and added references. - Unified notation used for the empty set symbol. - Added a clarification on the notation used for tensor powers. - Clarified connection to the ice rule - Added a new subsection clarifying the relation of the partition function to the trace formula