  Recently, SU(3) chains in the symmetric and self-conjugate representations
have been studied using field theory techniques. For certain representations,
namely rank-$p$ symmetric ones with $p$ not 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-$p$ symmetric representation with $n$ and $p$
coprime, we predict gapless excitations above the ground state. If $p$ is a
multiple of $n$, we predict a unique ground state with a finite energy gap.
Finally, if $p$ and $n$ have 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-Shultz-Mattis-Affleck theorem, and extend it to
SU($n$) models with longer range interactions.
