Higher rank elliptic partition functions and multisymmetric elliptic functions

TL;DR

This work constructs a unified family of $\mathfrak{gl}_{M+1}$ partition functions across rational, trigonometric, and elliptic regimes via a nested Izergin–Korepin approach, producing explicit multisymmetric weight functions that generalize both the Foda–Manabe framework and elliptic weight functions. By developing a layered lattice formalism with left/right quantum sites and leveraging a nested Korepin lemma, the authors derive exact, permutation-summed expressions $W$ that equal the partition functions $\psi$ in each regime, including dynamical parameters in the elliptic case. They further connect these multisymmetric functions to established weight-function frameworks (Konno; Rimányi–Tarasov–Varchenko), providing a coherent bridge between rational, trigonometric, and elliptic theories and paving the way for potential links to quiver varieties and Bethe–Gauge correspondences. Overall, the paper offers a unified, higher-rank extension of known partition-function/weight-function correspondences with explicit constructions across all three fundamental integrable regimes. The results enhance understanding of off-shell nested Bethe wavefunctions and their functional-analytic structure in higher rank, with potential applications to representation theory and mathematical physics.

Abstract

We introduce and investigate a class of $\mathfrak{gl}_{M+1}$ partition functions which is an extension of the one introduced by Foda-Manabe. We characterize the partition functions by a nested version of Izergin-Korepin analysis, and determine the explicit forms, for each of the rational, trigonometric and elliptic versions. The resulting multisymmetric functions can be regarded as extensions of the rational, trigonometric and elliptic weight functions.

Figures (17)

• Figure 1: Matrix elements of the rational $R$-matrix $R(x,y)$.
• Figure 2: The Yang-Baxter equation \ref{['rationalYBE']}.
• Figure 3: Rational and trigonometric $\mathfrak{gl}_{M+1}$ partition functions \ref{['rationalpartitionfunctions']} and \ref{['trigonometricpartitionfunctions']}. For the trigonometric case, the $x$- and $y$-variables are replaced by $u$- and $v$-variables.
• Figure 4: Pictorial explanation of the induced label.
• Figure 5: Sets used to describe the configurations. On the left panel, each row corresponds to each layer of the partition functions. The bottom row corresponds to the first layer where only colors 1 and 2 are allowed. The row above the bottom row corresponds to the second layer where color 3 is additionally allowed, and so on. ${\bm I}_{k_j}^{(j)}$ denote the set of coordinates which are colored by $1,2,\dots,j$ from the $j$-th to the $M$-th layer, which is represented as the thick line. The set of all coordinates from the $j$-th to the $M$-th layer is denoted as $\widehat{{\bm I}}_{k_j}^{(j)}$ and corresponds to the dashed line. The right panel represents the induced set $\widetilde{{\bm I}}_{k_j}^{(j)}$. The set obtained by removing from $\widehat{{\bm I}}_{k_j}^{(j)}$ coordinates colored by $j+2,\dots,M+1$ and squeezing to $\{1,2,\dots,k_{j+1}+L_j^{\mathrm{I}}+L_j^{\mathrm{I\!I}} \}$ is represented as the dashed line, and the subset ${\bm I}_{k_j}^{(j)}$ is accordingly mapped to some subset in $\{1,2,\dots, k_{j+1}+L_j^{\mathrm{I}}+L_j^{\mathrm{I\!I}} \}$ which is denoted as $\widetilde{{\bm I}}_{k_j}^{(j)}$ and corresponds to the thick line.

Theorems & Definitions (20)

• Proposition 2.1
• proof
• Definition 2.2
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Proposition 2.6