  In this paper we construct resolutions of finite dimensional irreducible
gl(m|n)-modules in terms of generalized Verma modules. The resolutions are
determined by the Kostant cohomology groups and extend the strong
(Lepowsky-)Bernstein-Gelfand-Gelfand resolutions to the setting of Lie
superalgebras. It is known that such resolutions for finite dimensional
representations of Lie superalgebras do not exist in general. Thus far they
have only been discovered for gl(m|n) in case the parabolic subalgebra has
reductive part equal to gl(m)+gl(n) and for tensor modules. In the current
paper we prove the existence of the resolutions for tensor modules of gl(m|n)
or sl(m|n) and their duals for an extensive class of parabolic subalgebras
including the ones already considered in the literature.
