Quiver semi-invariants and SAGBI bases

Abstract: We introduce a new combinatorial structure of linked tableaux, which generalize the semi-standard tableaux that index a SAGBI basis of the Pl\"ucker coordinate ring of a flag variety. We show that linked tableaux index Domokos-Zubkov semi-invariants, which generate the semi-invariant ring of a quiver, which in many cases coincides with the Cox ring of an associated quiver moduli space. We show that these semi-invariants satisfy straightening laws. In the case of the generalized Kronecker quiver, we prove that the semi-invariants associated to semi-standard linked tableaux are a (possibly infinite) SAGBI basis. For the generalized Kronecker quiver with dimension vector (2,2), we show that the SAGBI basis is finite and describe it explicitly.