Supersymmetric Grand Unified Theories (SGUTs) have achieved some degree of success, already present in the minimal models (with SU(5) or SO(10)). However, there are open problems that suggest the need to incorporate more elaborate constructions, spec
ifically the use of higher-dimensional representations in the Higgs sector. For example, a $45$ representation of SU(5) is often included to obtain correct mass relations for the first and second families of d-type quarks and leptons. When one adds these higher-dimensional Higgs representations one must verify the cancellation of anomalies associated to their fermionic partners. One possible choice, free of anomalies, include both $45,overline{45}$ representations to cancel anomalies. We review the necessary conditions for the cancellation of anomalies and discuss the different possibilities for supersymmetric SU(5) models. Alternative anomaly-free combinations of Higgs representations, beyond the usual vectorlike 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 multidimensional representations leads to different scenarios for the perturbative validity of the theory up to the Planck scale. We study the effect on the evolution of the gauge coupling up to the Planck scale due to the different sets of fields and representations that can render an anomaly-free model.
Top quark decays are of particular interest as a mean to test the standard model (SM) predictions, both for dominant (t -> b + W) and rare decays (t -> q + W, cV, cVV, c phi^{0}, bWZ). As the latter are highly suppressed, they become an excellent win
dow to probe the predictions of thories beyond the SM. In particular, we evaluate the corrections from new physics to the CKM-suppressed SM top quark decay t -> q + W (q = d, s), both within the an effective model with right-handed currents and the MSSM. We also discuss the perspectives to probe those predictions at the ILC.
