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

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 $U_q\left(gl{\left(2r\right|1)}^{\left(2\right)}\right)$ and the non-twisted quantum affine algebra $U_q\left(so{(2r+1)}^{\left(1\right)}\right)$, we proposed, in the previous paper [1], a Wronskian solution of the T-system for $U_q\left(so{(2r+1)}^{\left(1\right)}\right)$ as a reduction (folding) of the Wronskian solution for the non-twisted quantum affine superalgebra $U_q\left(gl{\left(2r\right|1)}^{\left(1\right)}\right)$. In this paper, we elaborate on this solution, and give a proof missing in [1]. In particular, we explain its connection to the Cherednik-Bazhanov-Reshetikhin (quantum Jacobi-Trudi) type determinant solution known in [2]. 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 $U_q\left(so{(2r+1)}^{\left(1\right)}\right)$ are related to reductions of T-functions for asymptotic typical representations of $U_q\left(gl{\left(2r\right|1)}^{\left(1\right)}\right)$.