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

TL;DR

This work investigates higher-rank multiple commutation relations in the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$ by a conceptual, diagrammatic method that identifies coefficients with specializations of trigonometric weight functions. The main result expresses products of L-operator entries as a structured sum over partitions, with coefficients given by $W$ and $H$, linking them to Izergin-Korepin determinants in rank one. It also leverages two universal Bethe-vector formalisms to construct a Gelfand-Tsetlin basis for the vector representation, clarifying the relationship between different GT realizations (Nazarov–Tarasov, Molev) and their constants. Parallel results for the Yangian $Y_h(\mathfrak{gl}_N)$ are established, and the rank-one case provides a constructive bridge to classical determinant expressions. The methods illuminate the interplay between quantum groups, nested Bethe vectors, and GT bases with potential applications to correlation functions and partition functions in integrable models.

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

Figures (19)

• Figure 1: Non-zero matrix elements of the trigonometric $R$-matrix.
• Figure 2: Graphical description of the Yang-Baxter equation.
• Figure 3: Graphical description of the unitarity relation.
• Figure 4: Graphical description of $T^{\mathrm{vect}}(u;\overline{\xi})$ and $T_{ij}^{\mathrm{vect}}(u;\overline{\xi})$.
• Figure 5: The partition function.

Theorems & Definitions (37)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Lemma 3.4
• proof