In this paper we give a new formula for the characters of finite-dimensional irreducible \(\mathfrak {gl}(m,n)\) -modules. We follow the same way as Su and Zhang did. First we give a new proof and a new formulation of the conjecture of Van der Jeugt, Hughes, King and Thierry-Mieg using weight and cap diagrams introduced by J. Brundan and C. Stroppel. Then we calculate the generating function of integer points of the corresponding polyhedron. Su and Zhang calculated that function by representing the polyhedron as the union of fundamental domains under the action of symmetric group. In our approach we calculate the generating function by means of Brion’s theorem.
