Homology of Littlewood complexes

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 $\mathbf{{S}}_{[\lambda ]}(V)$ of $\mathbf{{Sp}}(V)$. This representation admits a resolution by a natural complex $L^{\lambda }_{\bullet }$, which we call the Littlewood complex, whose terms are restrictions of representations of $\mathbf{{GL}}(V)$. When $\lambda$ has more than $n$ parts, the representation $\mathbf{{S}}_{[\lambda ]}(V)$ is not defined, but the Littlewood complex $L^{\lambda }_{\bullet }$ still makes sense. The purpose of this paper is to compute its homology. We find that either $L^{\lambda }_{\bullet }$ is acyclic or it has a unique nonzero homology group, which forms an irreducible representation of $\mathbf{{Sp}}(V)$. The nonzero 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 $\mathbf{{Sp}}(\infty )$ 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.