where X is a GUT singlet. Such superpotentials play important role in inflationary cos- mology, in the context of hybrid inflation at the GUT scale. In particular the most general Higgs sector (containing many scalar fields that could play constructive roles in inflation) can be constructed once the representations of all the fields present are known. For instance, SO (10) can contain Higgs field in many representations, such as a fundamental 10 H$_{i}$ , a four index 210 field Φ$_{ijkl}$, or a 3 index 120 field Ω$_{ijk}$. It is possible to construct a mass term, for example Φ 2 symbolizing, Φ$_{ijkl}$Φ$_{ijkl}$, but it is not possible to construct a scalar term (with all indices contracted) using three factors of H , or Ω 41 . For example, in the minimal SO (10) [441], the field content which realize the breaking of SO (10) are Φ$_{ijkl}$ in 210 , H$_{i}$ in 210 , Σ$_{ijklm}$ and ¯ Σ$_{ijklm}$ in 126 and 126 . The most general Higgs superpotential is then given by $^{42}$: W$_{H}$ = m$_{Φ}$ Φ 2 + λ$_{Φ}$ Φ 3 + m$_{H}$H 2 + m$_{Σ}$ ΣΣ + η ΦΣΣ + Φ H ( α Σ + α Σ) (155) The possible symmetry breaking depends on the minima of the full superpotential, but only MSSM singlets inside each Higgs fields can take a non-vanishing VEV. For example, in the minimal SO (10), there are three MSSM singlets inside Φ. The decomposition is given by: 210 = ( 15 , 1 , 1 ) + ( 1 , 1 , 1 ) + ( 15 , 1 , 3 ) + ( 15 , 3 , 1 ) + ( 6 , 2 , 2 ) + ( 10 , 2 , 2 ) + ( 10 , 2 , 2 ) under the Pati-Salam subgroup SU (4)$_{c}$ × SU (2)$_{L}$ × SU (2)$_{R}$ [430]. The MSSM singlets are found in (1 , 1 , 1), (15 , 1 , 1) and (15 , 1 , 3). The minimal SUSY SO (10) model contains 2 additional MSSM singlets in Σ and Σ. Candidates for the inflaton should be searched for within these components. All the minima can be found by solving for the values of the singlet fields minimizing all the F -terms. The symmetries of these minima are given by the number of invariant generators of SO (10). 41 The same conclusion can be made by looking at whether the product of three 10 contains a singlet of SO (10) or not. 42 The fermionic sector is also written in the form of a superpotential. The only Higgs coupling to the fermions generate Yukawa terms are in a representation R such that 16 × 16 × R contains a singlet. In the case of a minimal SO (10), this is the case only for H and Σ and the superpotential generating the fermion mass matrix: W$_{F}$ = Ψ$_{a}$( Y$_{H}$H + Y$_{Σ}$ Σ)Ψ$_{b}$, where a,b are family indices and the Y are the symmetric Yukawa matrices. The most general fermionic sector contains only one additional term involving a 120 as 16 × 16 × 120 . One can also enlarge the minimal SO (10) model to have a more realistic fermionic mass matrices [442, 447, 448]. 70