On the orthogonal dimension of orbital sets

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(alpha)* = e(alpha(1)) *...* e(alpha(m)), e(beta)* = e(beta(1)) *...* e(beta(m)) is an element of V-lambda (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 alpha, beta) more manageable conditions of orthogonality of e(alpha)* and e(beta)*. 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-lambda(S-m), extracted from {e(alpha)* : alpha is an element of Gamma(m,n)} is concluded. (c) 2003 Elsevier Inc. All rights reserved.