About Group Representations in Grand Unified Theories

The main purpose of this article is to systematize group theory methods in Grand Unified Theories (GUTs). An exact representation structure for all admissible groups that unify elementary forces is analysed. The implications of requirements asymptotic freedom for standard SU (3)_c ⊗ SU(2)_L ⊗ U(1)_y model is investigated in detail. It is shown how restrictive are such assumptions in unified model building. As a result the maximal number of families is five independently on the kind of considered model. In several cases this number is even smaller. The maximal horizontal structure of interactions which is possible to embed into unification scheme is also discussed for all admissible models. The number of superheavy particles is evaluated in different unifying schemes.