15.17.5 Tensor Symmetric Type Preservation

Tensor Symmetric Type Preservation is the generalization of transformation-preservation results beyond the single case of full symmetry, addressing every possible symmetry type a tensor can carry, as classified by partitions of its order, and identifying the precise circumstances, tied to the characteristic of the underlying field, under which a tensor's symmetry type is guaranteed to survive a change of basis.

Symmetry Types Beyond Full Symmetry

Classification by Partition Shape

A tensor of order d need not be either fully symmetric or fully antisymmetric; more generally, its behavior under the permutation action of the symmetric group on d letters can match any of the irreducible representations of that group, and these irreducible representations are classified by partitions of d, equivalently by Young diagrams with d boxes. Full symmetry corresponds to the partition with a single part of size d, full antisymmetry corresponds to the partition consisting of d parts each of size one, and every partition in between corresponds to a mixed symmetry type, in which the tensor is symmetric under some subsets of index permutations and antisymmetric or unconstrained under others.

Type as an Intrinsic Label

The symmetry type of a tensor, in this classification sense, is the label identifying which irreducible representation (or, for tensors that are not pure isotypic vectors, which combination of irreducible representations) the tensor spans under the permutation action. Type Preservation addresses whether this label remains attached to the tensor after a change of basis is applied to the underlying vector space.

The Preservation Statement in Characteristic Zero

Commuting Actions Guarantee Preservation

As established under Subspace Invariance, the action of the general linear group on tensor space commutes with the action of the symmetric group permuting tensor factors. In characteristic zero, or more generally whenever the order d is invertible in the underlying field, this commutativity, combined with the complete reducibility of the symmetric group's representations (Maschke's theorem applied to the group algebra of the symmetric group), guarantees that every isotypic component associated with a partition is a well-defined, general-linear-group-invariant subspace, and a tensor lying purely in one isotypic component before a change of basis continues to lie purely in that same component afterward. This is Type Preservation in its clean, unconditional form, generalizing the single-type statement of Transformation Preservation for full symmetry to every symmetry type simultaneously.

Explicit Verification for Small Order

For order two, the two types (symmetric and antisymmetric) are preserved because the symmetrizing and antisymmetrizing projectors, built by averaging over the two-element symmetric group with the coefficient one half, commute with any linear change of basis applied identically to both tensor factors; the coefficient one half is well-defined precisely because two is invertible whenever the characteristic is not two, foreshadowing the characteristic-dependent subtlety addressed below.

Breakdown in Small Characteristic

The Role of Characteristic Dividing the Order

The averaging projectors used to extract a tensor's component of a given symmetry type involve dividing by the order of the symmetric group or by related combinatorial factors such as factorials of the partition parts; when the characteristic of the underlying field divides one of these factors, in particular whenever it divides d factorial, the relevant projector is no longer well-defined, and the clean decomposition of tensor space into isotypic pieces used to state Type Preservation can fail to exist in the same form. This is the same characteristic obstruction encountered in the Tensor Quadratic Form Polarization Relation, where division by two failed in characteristic two, now recognized as a special case of a general phenomenon affecting symmetry-type decompositions of every order.

Modular Representation Theory Consequences

In this small-characteristic setting, the representation theory of the symmetric group is no longer semisimple, and distinct symmetry types can become linked in ways that have no counterpart in characteristic zero: a tensor that appears to have one symmetry type in one basis may, after a change of basis, produce a tensor whose type is only partially determined, or that lies in a larger, non-split extension of types rather than in a single isotypic piece. Type Preservation, in its strong form, is therefore restricted in scope to fields of characteristic zero or of characteristic exceeding the order d, and the boundary cases are handled instead by the specialized methods of modular representation theory rather than by the direct argument used for symmetric tensors throughout the rest of this material.

Significance Within the Broader Theory

Justifying the Restriction to Characteristic Zero

Type Preservation is the structural reason that Tensor Symmetric Decomposition Structure, apolarity, and the Alexander-Hirschowitz classification are conventionally developed over fields of characteristic zero, most often the complex or real numbers: only in this setting is the symmetric subspace guaranteed to remain cleanly and permanently separated, under every change of basis, from tensors of other symmetry types, ensuring that a tensor identified as symmetric in one coordinate system cannot be mistaken, after a change of basis, for a tensor of mixed or antisymmetric type.

Relevance to Applications Over Finite Fields

In applications involving finite fields, such as coding theory or combinatorics, where the characteristic may indeed divide the order of the tensors under study, Type Preservation failures are a genuine practical concern, and algorithms relying on symmetrization or type-based decomposition must verify separately that the relevant characteristic obstruction does not arise before applying results developed under the standing assumption of characteristic zero.