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

TL;DR

This paper proves a saturation theorem for special orthogonal and symplectic groups, extending previous results and employing quiver invariant theory and representation techniques.

Contribution

It establishes a new saturation property for classical groups using quiver invariant theory and representation methods, extending prior work in the field.

Findings

Proves saturation for orthogonal and symplectic groups.

Extends previous saturation results to new group classes.

Uses quiver invariant theory and generic representation theory.

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