On the orthogonal dimension of orbital sets Original Research Article

Let V be an inner product vector space over image and (e1,…,en) an orthonormal basis of V. A combinatorial necessary and sufficient condition for orthogonality of critical decomposable symmetrized tensorsimagee*α=eα(1)*cdots, three dots, centered*eα(m),e*β=eβ(1)*cdots, three dots, centered*eβ(m)set membership, variantVλ(Sm)with “factors” extracted from (e1,…,en) 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 Sm. From this computation, the nonexistence of orthogonal bases of Vλ(Sm), extracted from {eα*:αset membership, variantΓm,n}, is concluded.