On the orthogonal dimension of orbital sets

Abstract

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.

Symmetry Classes of Polynomials

“…Thus it is sufficient to show that there is ∈¯ + , such that the orbital subspace V * of V S m is critical and does not have orthogonal basis. In view of Corollaries 1 through 4 of [3], these all follow from the following observation.…”

“…In [3,14], for some irreducible characters of S m , the nonexistence of orthogonal bases of V S m , extracted from e * ∈ 0 m d , is concluded. Now we obtain the similar result for the symmetry classes of polynomials.…”

Orthogonal bases of Brauer symmetry classes of tensors for groups having cyclic support on non-linear Brauer characters

“…During the past decades, there are many papers devoted to study symmetry classes of tensors, see, for example, [1][2][3][4][5][6][7][8][9]. One of the active research topics is the investigation of the special basis (o-basis) for the classes.…”

Symmetry classes of tensors associated with the semi-dihedral groups SD8n

“…If χ is not linear, it is possible that V χ (G) has no orthogonal * -basis. The reader can find further information about the symmetry classes of tensors in [1][2][3][4][5][6][7][8], [11][12][13][14][15] and [17]. In this paper we discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with semi-dihedral groups SD 8n .…”