  In this paper we construct free resolutions of certain class of closed
subvarieties of affine spaces (the so-called "opposite big cells" of
Grassmannians). Our class covers the determinantal varieties, whose resolutions
were first constructed by A. Lascoux (Adv. Math., 1978). Our approach uses the
geometry of Schubert varieties. An interesting aspect of our work is its
connection to the computation of the cohomology of homogeneous bundles (that
are not necessarily completely reducible) on partial flag varieties.
