A toric degeneration of Kronecker moduli spaces

In this paper, we show that there is a finite SAGBI basis of the coordinate ring of a Kronecker quiver moduli space, indexed by primitive semi-standard tableaux pairs. This induces a toric degeneration of the Kronecker moduli space to a normal toric variety, a generalization of the toric degeneration of the Grassmannian to the Gelfand–Cetlin polytope constructed by Gonciulea–Lakshmibai. The moment polytope of the degenerate toric variety can be described as the intersection of two Gelfand–Cetlin polytopes. We explain when this can be generalized to degenerations coming from matching fields.