  We consider the methods by which higher-level and non-simply laced gauge
symmetries can be realized in free-field heterotic string theory. We show that
all such realizations have a common underlying feature, namely a dimensional
truncation of the charge lattice, and we identify such dimensional truncations
with certain irregular embeddings of higher-level and non-simply laced gauge
groups within level-one simply-laced gauge groups. This identification allows
us to formulate a direct mapping between a given subgroup embedding, and the
sorts of GSO constraints that are necessary in order to realize the embedding
in string theory. This also allows us to determine a number of useful
constraints that generally affect string GUT model-building. For example, most
string GUT realizations of higher-level gauge symmetries G_k employ the
so-called diagonal embeddings G_k\subset G\times G \times...\times G. We find
that there exist interesting alternative embeddings by which such groups can be
realized at higher levels, and we derive a complete list of all possibilities
for the GUT groups SU(5), SU(6), SO(10), and E_6 at levels k=2,3,4 (and in some
cases up to k=7). We find that these new embeddings are always more efficient
and require less central charge than the diagonal embeddings which have
traditionally been employed. As a byproduct, we also prove that it is
impossible to realize SO(10) at levels k>4. This implies, in particular, that
free-field heterotic string models can never give a massless 126 representation
of SO(10).
