Seminar

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 $\bS_{[\lambda]}(V)$ of $\Sp(V)$. This representation admits a resolution by a natural complex $L^{\lambda}_{\bullet}$, (called {\bf Littlewood complex}), whose terms are restrictions of representations of $\GL(V)$. When $\lambda$ has more than $n$ parts, the representation $\bS_{[\lambda]}(V)$ is not defined, but the Littlewood complex $L^{\lambda}_{\bullet}$ still makes sense.

I will explain how to compute its homology. One finds that either $L^{\lambda}_{\bullet}$ is acyclic or 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 categorifies earlier results of Koike--Terada on universal character rings. The result has an interpretation in terms of commutative algebra: it calculates the Koszul homology of the ideal generated by invariants of the symplectic group on the set of vectors.

One can prove analogous results for orthogonal and general linear groups.

If time permits I will show how some of these ideas can be generalized to exceptional groups.

The talk is based on the joint paper with Andrew Snowden and Steven Sam and another one with Steven Sam.