  One interesting combinatorial feature of classical determinantal varieties is
that the character of their coordinate rings give a natural truncation of the
Cauchy identity in the theory of symmetric functions. Natural generalizations
of these varieties exist and have been studied for the other classical groups.
In this paper we develop the relevant properties from scratch. By studying the
isotypic decomposition of their minimal free resolutions one can recover
classical identities due to Littlewood for expressing an irreducible character
of a classical group in terms of Schur functions. We propose generalizations
for the exceptional groups. In type G_2, we completely analyze the variety and
its minimal free resolution and get an analogue of Littlewood's identities. We
have partial results for the other cases. In particular, these varieties are
always normal with rational singularities.
