Standard Majorana representations of the symmetric groups

Let G be a finite group, W be a R[G]-module equipped with a G-invariant positive definite bilinear form (,)_W, and X a finite generating set of W such that X is transitively permuted by G. We show a new method for computing the dimensions of the irreducible constituents of W. Further, we apply this method to Majorana representations of the symmetric groups and prove that the symmetric group S_n has a Majorana representation in which every permutation of type (2, 2) of S_n corresponds to a Majorana axis if and only if n≤12.