Littlewood Complexes and Analogues of Determinantal Varieties

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 analog of Littlewood's identities. We have partial results for the other cases. In particular, these varieties are always normal with rational singularities.