Recently, SU(3) chains in the symmetric and self-conjugate representations have been studied using field theory techniques. For certain representations, namely rank-psymmetric ones with pnot a multiple of 3, it was argued that the ground state exhibits gapless excitations. For the remaining representations considered, a finite energy gap exists above the ground state. In this paper, we extend these results to SU(n) chains in the symmetric representation. For a rank-psymmetric representation with nand pcoprime, we predict gapless excitations above the ground state. If pis a multiple of n, we predict a unique ground state with a finite energy gap. Finally, if pand nhave a greatest common divisor 1 < q< n, we predict a ground state degeneracy of n/q, with a finite energy gap. To arrive at these results, we derive a non-Lorentz invariant flag manifold sigma model description of the SU(n) chains, and use the renormalization group to show that Lorentz invariance is restored at low energies. We then make use of recently developed anomaly matching conditions for these Lorentz-invariant models. We also review the Lieb-Schultz-Mattis-Affleck theorem, and extend it to SU(n) models with longer range interactions. (C) 2020 The Author(s). Published by Elsevier B.V.