Littlewood complexes and analogues of determinantal varieties

TL;DR

This paper explores generalizations of determinantal varieties related to classical and exceptional groups, revealing new identities for characters and analyzing their geometric properties.

Contribution

It develops properties of generalized varieties for classical and exceptional groups, deriving identities and analyzing their resolutions from scratch.

Findings

Classical identities are recovered via isotypic decompositions.

Type G_2 varieties are fully analyzed with minimal free resolutions.

Generalized varieties are shown to be normal with rational singularities.

Abstract

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.