# Mathematical Sciences Research Institute

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# Seminar

Koszul homology of ideals generated by invariants and analogues of determinantal varieties February 04, 2013 (02:00pm PST - 03:00pm PST)
Location: MSRI: Simons Auditorium
Speaker(s) Jerzy Weyman (Northeastern University)
Description No Description

Video
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.