Homology of Littlewood complexes

TL;DR

This paper computes the homology of Littlewood complexes associated with symplectic, orthogonal, and general linear groups, revealing when they are acyclic or have a single non-zero homology, thus extending representation theory.

Contribution

It provides explicit homology computations for Littlewood complexes beyond classical cases, generalizing the Borel-Weil-Bott theorem and categorifying earlier universal character results.

Findings

Littlewood complexes are either acyclic or have one non-zero homology group.

The non-zero homology can be computed using a rule similar to Borel-Weil-Bott.

Results apply to symplectic, orthogonal, and general linear groups.

Abstract

Let V be a symplectic vector space of dimension 2n. Given a partition \lambda with at most n parts, there is an associated irreducible representation S_{[\lambda]}(V) of Sp(V). This representation admits a resolution by a natural complex L^\lambda, which we call the Littlewood complex, whose terms are restrictions of representations of GL(V). When \lambda has more than n parts, the representation S_{[\lambda]}(V) is not defined, but the Littlewood complex L^\lambda still makes sense. The purpose of this paper is to compute its homology. We find that either L^\lambda is acyclic or that it has a unique non-zero homology group, which forms an irreducible representation of Sp(V). The non-zero homology group, if it exists, can be computed by a rule reminiscent of that occurring in the Borel-Weil-Bott theorem. This result can be interpreted as the computation of the "derived specialization" of irreducible representations of Sp(\infty), and as such categorifies earlier results of Koike-Terada on universal character rings. We prove analogous results for orthogonal and general linear groups. Along the way, we will see two topics from commutative algebra: the minimal free resolutions of determinantal ideals and Koszul homology.