Skew Invariant Theory of Symplectic Groups, Pluri-Hodge Groups and 3-Manifold Invariants

Abstract

The first has to do with skew invariant theory. Let V be a finite dimensional complex vector space with a skew symmetric non-degenerate form and let Sp(V ) be the corresponding symplectic group. For a complex vector space W we consider the action of Sp(V ) on the exterior algebra Λ(V ⊗W ) (the action being trivial on W ). In Theorem 2.3 we give generators and relations for the algebra of Sp(V ) invariants for this action (“Fundamental theorem for skew invariant theory”). The relations are the so-called “Pn” relations which appear in the study of certain 3-manifold invariants. Next we show that there is a natural action of Sp(g), where g = dimC W , on Λ(V ⊗W ), by considering the spin representation of the orthogonal algebra associated to V ⊗ (W ⊕ W ∗). The action of Sp(V )× Sp(g) on Λ(V ⊗W ) is multiplicity free and we determine the highest weights of the representations which occur, Theorem 2.4.