Symmetric quivers, invariant theory, and saturation theorems for the classical groups

Abstract

Let G denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for G: given dominant weights λ1,…,λr such that the tensor product VNλ1⊗⋯⊗VNλr contains nonzero G-invariants for some N⩾1, we show that the tensor product V2λ1⊗⋯⊗V2λr also contains nonzero G-invariants. This extends results of Kapovich–Millson and Belkale–Kumar and complements similar results for the general linear group due to Knutson–Tao and Derksen–Weyman. Our techniques involve the invariant theory of quivers equipped with an involution and the generic representation theory of certain quivers with relations.