Abstract. Let g ′ and g be isomorphic to any two of the Lie algebras gl(∞), sl(∞), sp(∞), and so(∞). Let M be a simple tensor g-module ([PSt], [PSe]). We introduce the notion of an embedding g ′ ⊂ g of general tensor type and derive branching laws for triples g ′ , g, M , where g ′ ⊂ g is an embedding of general tensor type. More precisely, since M is in general not semisimple as a g ′ -module, we determine the socle filtration of M over g ′ . Due to the description of embeddings of classical locally finite Lie algebras given in [DP], our results hold for all possible embeddings g ′ ⊂ g unless g ′ ∼ = gl(∞).
Let K be a connected compact semisimple group and V λ be an irreducible unitary representation with highest weight λ. We study the momentum map µ : P(V λ ) → k * . The intersection µ(P) + = µ(P) ∩ t + of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of V λ . We construct an affine rational polyhedral convex cone Υ λ with vertex λ, such that µ(P) + ⊂ Υ λ ∩ t + . We show that equality holds for a class of representations, including those with regular highest weight. For those cases, we obtain a complete combinatorial description of the momentum polytope, in terms of λ. We also present some results on the critical points of ||µ|| 2 . Namely, we consider the existence problem for critical points in the preimages of Kirwan's candidates for critical values. Also, we consider the secant varieties to the unique complex orbit X ⊂ P, and prove a relation between the momentum images of the secant varieties and the degrees of K-invariant polynomials on V λ .
In this paper, we consider the exterior algebra Λ(W ) of a polynomial GL(n)-module W and use previously developed methods to determine the Hilbert series of the algebra of invariants Λ(W ) G , where G is one of the classical complex subgroups of GL(n), namely SL(n), O(n), SO(n), or Sp(2d) (for n = 2d). Since Λ(W ) G is finite dimensional, we apply the described method to compute a lot of explicit examples. For Λ(S 3 C 3 ) SL(3) , using the computed Hilbert series, we obtain an explicit set of generators.2010 Mathematics Subject Classification. 13A50; 15A72; 15A75; 20G05.
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