In the framework of supersymmetric Grand Unified Theories, the minimal Higgs sector is often extended by introducing multi-dimensional Higgs representations in order to obtain realistic models. However these constructions should remain anomaly-free,
which constraints significantly their structure. We review the necessary conditions for the cancellation of anomalies in general and discuss in detail the different possibilities for SUSY SU(5) models. Alternative anomaly free combinations of Higgs representations, beyond the usual vector-like choice, are identified, and it is shown that their corresponding beta functions are not equivalent. Although the unification of gauge couplings is not affected, the introduction of multi-dimensional representations leads to different scenarios for the perturbative validity of the theory up to the Planck scale.
