On the orthogonal dimension of orbital sets

The concept of orthogonal dimension of a finite set of nonzero vectors is introduced. Using the above mentioned condition, the orthogonal dimension of critical orbital sets is computed for a class of irreducible characters of Sm. From this computation, the nonexistence of orthogonal bases of Vλ(Sm), extracted from {eα*:α∈Γm,n}, is concluded.