Quantum Jacobi-Trudi and Giambelli Formulae for $U_q(B_r^{(1)})$ from Analytic Bethe Ansatz

TL;DR

This paper derives quantum analogues of classical Jacobi-Trudi and Giambelli formulae for $U_q(B_r^{(1)})$ modules, providing explicit solutions to the $T$-system functional relation in integrable models.

Contribution

It introduces quantum Jacobi-Trudi and Giambelli formulae for $U_q(B_r^{(1)})$, extending classical symmetric function identities to the quantum affine algebra context.

Findings

Derived explicit formulae for transfer matrix spectra.

Established quantum analogues of classical symmetric function identities.

Provided a full solution to the $T$-system functional relation.

Abstract

Analytic Bethe ansatz is executed for a wide class of finite dimensional $U_q(B^{(1)}_r)$ modules. They are labeled by skew-Young diagrams which, in general, contain a fragment corresponding to the spin representation. For the transfer matrix spectra of the relevant vertex models, we establish a number of formulae, which are $U_q(B^{(1)}_r)$ analogues of the classical ones due to Jacobi-Trudi and Giambelli on Schur functions. They yield a full solution to the previously proposed functional relation ($T$-system), which is a Toda equation