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

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.