  The irreducible representations of complex semisimple algebraic groups with
finitely many orbits are parametrized by graded simple Lie algebras. For the
exceptional Lie algebras, Kraskiewicz and Weyman exhibit the Hilbert
polynomials and the expected minimal free resolutions of the normalization of
the orbit closures. We present an interactive method to construct explicitly
these and related resolutions in Macaulay2. The method is then used in the
cases of the Lie algebras of type $E_6$, $F_4$ and $G_2$, to confirm the shape
of the expected resolutions as well as some geometric properties of the orbit
closures.

Representation Theory;Commutative Algebra;Algebraic Geometry