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

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 $E_{τ,η}(gl_3)$ to introduce an elliptic analogue of the partition functions associated with $E_{τ,η}(gl_3)$. 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 Rimányi-Tarasov-Varchenko, Konno, Felder-Rimányi-Varchenko.

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