Analytic Bethe ansatz and functional equations associated with any simple root systems of the Lie superalgebra sl(r+1 s+1)

The Lie superalgebra sl(r+1 s+1) admits several inequivalent choices of simple root systems. We have carried out analytic Bethe ansatz for any simple root systems of sl(r+1 s+1). We present transfer matrix eigenvalue formulae in dressed vacuum form, which are expressed as the Young supertableaux with some semistandard-like conditions. These formulae have determinant expressions, which can be viewed as quantum analogue of Jacobi–Trudi and Giambelli formulae for sl(r+1 s+1). We also propose a class of transfer matrix functional relations, which is specialization of Hirota bilinear difference equation. Using the particle–hole transformation, relations among the Bethe ansatz equations for various kinds of simple root systems are discussed.