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 . We show that all the coefficients in the multiple commutation relations between the -operator elements are given in terms of the trigonometric weight functions for the vector representation, independent of the representation of the -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 -operator elements from the constructions by Nazarov-Tarasov or Molev. We also present corresponding results for the Yangian .
Keywords
Cite
@article{arxiv.2510.21233,
title = {Multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$, nested Bethe vector and the Gelfand-Tsetlin basis},
author = {Allan John Gerrard and Kohei Motegi and Kazumitsu Sakai},
journal= {arXiv preprint arXiv:2510.21233},
year = {2026}
}
Comments
44 pages, 23 figures. Ann. Henri Poincar\'e (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