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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Abstract. We approach Riordan arrays and their generalizations via umbral symbolic methods. This new approach allows us to derive fundamental aspects of the theory of Riordan arrays as immediate consequences of the umbral version of the classical Abel's identity for polynomials. In particular, we obtain a novel non-recursive formula for Riordan arrays and derive, from this new formula, some known recurrences and a new recurrence relation for Riordan arrays.

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