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Let V be an inner product vector space over C and (e 1 , . . . , e n ) an orthonormal basis of V . A combinatorial necessary and sufficient condition for orthogonality of critical decomposable symmetrized tensors e * α = e α(1) * · · · * e α(m) , e * β = e β(1) * · · · * e β(m) ∈ V λ (S m ) with "factors" extracted from (e 1 , . . . , e n ) is proved.The notion of sign-uniform partition is introduced and the set of the sign-uniform partitions is described.The characterization of the sign-uniform partitions is used to produce (for a class of pairs of congruent α, β) more manageable conditions of orthogonality of e * α and e * β . 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 S m . From this computation, the nonexistence of orthogonal bases of V λ (S m ), extracted from {e * α : α ∈ m,n }, is concluded.

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