A class of partition functions associated with $E_{\tau,\eta}(gl_3)$ by Izergin-Korepin analysis

TL;DR

This paper introduces an elliptic analogue of partition functions linked to the elliptic quantum group $E_{ au, ext{eta}}(gl_3)$, using a nested elliptic Izergin-Korepin analysis to derive explicit symmetrized forms.

Contribution

It develops a nested elliptic Izergin-Korepin analysis to explicitly formulate partition functions associated with $E_{ au, ext{eta}}(gl_3)$, connecting to known elliptic weight functions.

Findings

Derived explicit symmetrized forms of elliptic partition functions.

Connected new elliptic functions to previously studied elliptic weights.

Extended the analysis of integrable models to elliptic quantum groups.

Abstract

Recently, a class of partition functions associated with higher rank rational and trigonometric integrable models were introduced by Foda and Manabe. We use the dynamical $R$-matrix of the elliptic quantum group $E_{\tau,\eta}(gl_3)$ to introduce an elliptic analogue of the partition functions associated with $E_{\tau,\eta}(gl_3)$. We investigate the partition functions of Foda-Manabe type by developing a nested version of the elliptic Izergin-Korepin analysis, and present the explicit forms as symmetrization of multivariable elliptic functions. We show that special cases are essentially the elliptic weights functions introduced in the works by Rim\'anyi-Tarasov-Varchenko, Konno, Felder-Rim\'anyi-Varchenko.