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 Uq(gl(2r|1)(2)) and the non-twisted quantum affine algebra Uq(so(2r+1)(1)), we proposed, in the previous paper [1], a Wronskian solution of the T-system for Uq(so(2r+1)(1)) as a reduction (folding) of the Wronskian solution for the non-twisted quantum affine superalgebra Uq(gl(2r|1)(1)). 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 Uq(so(2r+1)(1)) are related to reductions of T-functions for asymptotic typical representations of Uq(gl(2r|1)(1)).