Boson-Fermion correspondence, QQ-relations and Wronskian solutions of the T-system

TL;DR

This paper develops and proves a Wronskian solution for the T-system of quantum affine algebra $U_q(so(2r+1)^{(1)})$, connecting superalgebra representations with non-graded algebra solutions and exploring their determinant and character formulas.

Contribution

It provides a detailed proof of a Wronskian solution for the T-system, linking it to quantum Jacobi-Trudi determinants and Weyl character formulas, and relates spinorial and typical representations.

Findings

Established a Wronskian solution for the T-system of $U_q(so(2r+1)^{(1)})$

Connected T-functions to quantum Jacobi-Trudi and Weyl character formulas

Linked spinorial T-functions to reductions of typical representation T-functions

Abstract

It is known that there is a correspondence between representations of superalgebras and ordinary (non-graded) algebras. Keeping in mind this type of correspondence between the twisted quantum affine superalgebra $U_{q}(gl(2r|1)^{(2)})$ and the non-twisted quantum affine algebra $U_{q}(so(2r+1)^{(1)})$, we proposed, in the previous paper [arXiv:1109.5524], a Wronskian solution of the T-system for $U_{q}(so(2r+1)^{(1)})$ as a reduction (folding) of the Wronskian solution for the non-twisted quantum affine superalgebra $U_{q}(gl(2r|1)^{(1)})$. In this paper, we elaborate on this solution, and give a proof missing in [arXiv:1109.5524]. In particular, we explain its connection to the Cherednik-Bazhanov-Reshetikhin (quantum Jacobi-Trudi) type determinant solution known in [arXiv:hep-th/9506167]. We also propose Wronskian-type expressions of T-functions (eigenvalues of transfer matrices) labeled by non-rectangular Young diagrams, which are quantum affine algebra analogues of the Weyl character formula for $so(2r+1)$. We show that T-functions for spinorial representations of $U_{q}(so(2r+1)^{(1)})$ are related to reductions of T-functions for asymptotic typical representations of $U_{q}(gl(2r|1)^{(1)})$.