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Unification of matter fields implies the existence of accidental permutationsymmetries, which potentially remain immune to large quantum corrections up tothe TeV scale. We investigate the case of a supersymmetric $SU(5)$ grandunified theory, where such a permutation symmetry is present in the up-typesquark sector. We present a variety of tests allowing to challenge the $SU(5)$hypothesis based on the observation of squarks at the LHC. These tests appearas relations among observables involving flavour-violating orchirality-flipping decays of squarks. Moreover, they rely on top-polarimetryand charm-tagging. As an example, we discuss the application to the scenario ofNatural Supersymmetry, while more examples can be found in the related journalpublications.

Accidental permutation symmetries as a test for Grand Unification: the supersymmetric $SU(5)$ case