  The problem of computing the characters of the finite dimensional irreducible
representations of the Lie superalgebra $\mathfrak{gl}(m|n)$ over $\C$ was
solved a few years ago by V. Serganova. In this article, we present an entirely
different approach. One consequence is a direct and elementary proof of a
conjecture made by van der Jeugt and Zhang for the composition multiplicities
of Kac modules. This does not seem to follow easily from Serganova's formula,
since that involves certain alternating sums. We also compute Ext's between
simple modules in the category of finite dimensional representations, and
formulate for the first time a conjecture for the characters of the infinite
dimensional irreducible representations in the analogue of category $\mathcal
O$ for the Lie superalgebra $\mathfrak{gl}(m|n)$.
