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

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 [1], 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 [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 (so(2r+1) (1) ) are related to reductions of T-functions for asymptotic typical representations of U q (gl(2r|1) (1) ).