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 λ^1, ..., λ^r such that the tensor product V_{Nλ^1} \otimes ... \otimes V_{Nλ^r} contains nonzero G-invariants for some N \ge 1, we show that the tensor product V_{2λ^1} \otimes ... \otimes V_{2λ^r} 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.

29 pages, no figures; v2: updated Theorem 2.4 to odd characteristic, added Remark 3.9, added references, corrected some definitions and typos