  A recent analysis of the methods by which higher-level gauge symmetries are
realized in string theory has made it possible to systematically obtain new
constraints on string GUT model-building. In this paper, we perform such a
study for the case of SO(10) free-field string GUT models, and find a number of
significant new results. First, we show that within the conventional string
models using the so-called ``diagonal'' embeddings, all representations larger
than the 16 must always transform as singlets under all gauge symmetries beyond
SO(10). This includes the 45 and 54 representations, and holds regardless of
the affine level at which SO(10) is realized. Furthermore, we show that such
constructions can never give rise to the 120, 126, or 144 representations of
SO(10) --- again regardless of the affine level. We also perform a similar
analysis for E_6, and find that the adjoint 78 representation must always
transform as a singlet under all gauge symmetries beyond E_6; moreover, all
representations beyond the 78 representation are prohibited. Taken together,
these results thus severely limit the types of phenomenologically realistic
field-theoretic SO(10) and E_6 models that can be obtained using such string
constructions. Finally, we also explore the possibility of using non-diagonal
free-field embeddings, and show that in the SO(10) case such embeddings may
have improved phenomenological prospects.
