15.17.4 Tensor Symmetric Subspace Invariance

Tensor Symmetric Subspace Invariance is the structural, representation-theoretic framing of why symmetric tensors form a distinguished linear subspace of the full tensor product space that is left globally unchanged by every linear transformation of the underlying vector space, situating this fact within the broader decomposition of tensor space into pieces indexed by the ways permutation symmetry can be distributed among the indices.

The Symmetric Subspace as a Fixed-Point Set

Definition via the Permutation Action

The symmetric group on d letters acts on the full tensor product space V tensor d by permuting the d tensor factors. The subspace of symmetric tensors is exactly the fixed-point set of this action: the set of tensors left unchanged by every permutation. Because the fixed points of any linear group action always form a linear subspace (a sum or scalar multiple of fixed vectors is again fixed), the symmetric tensors automatically form a subspace of V tensor d, without any additional argument beyond the general theory of group actions on vector spaces.

Two Commuting Actions

Separately from the permutation action of the symmetric group, the general linear group of V acts on V tensor d by applying a single linear transformation simultaneously to every tensor factor. These two actions, of the symmetric group and of the general linear group, commute with one another: permuting factors and then transforming each factor gives the same result as transforming each factor and then permuting them. This commutativity is the precise algebraic reason that the fixed-point subspace of the permutation action, namely the symmetric tensors, is automatically invariant under the separate action of the general linear group, since a general linear group element cannot move a fixed point of the permutation action out of the fixed-point set.

Consequences of the Two Commuting Actions

Subspace Invariance as a Structural Theorem

Restated in this language, Subspace Invariance says that the general linear group's action on V tensor d restricts to a well-defined action on the symmetric subspace, rather than mapping symmetric tensors outside of that subspace; this is the same conclusion reached by the direct component-level proof given under Transformation Preservation, but arrived at here purely from the commutativity of the two group actions, without reference to explicit indices.

Schur-Weyl Duality Context

The commuting actions of the symmetric group and the general linear group on tensor power spaces are the subject of Schur-Weyl duality, which decomposes the full tensor product space into isotypic pieces, each associated with a partition of d, on which the two groups act through dual, mutually determining representations. The symmetric tensors correspond to the piece associated with the partition consisting of the single part d (the trivial representation of the symmetric group), and the invariance of this piece under the general linear group is the smallest, most elementary instance of the general fact that every isotypic piece in the Schur-Weyl decomposition is separately invariant under the general linear group action.

The Complementary Pieces

The Antisymmetric Subspace

At the opposite extreme from the symmetric subspace lies the antisymmetric (alternating) subspace, the fixed points of the sign-twisted permutation action, corresponding to the partition of d into d single parts; this subspace is likewise invariant under the general linear group, and for order two it is exactly the complement used in the decomposition of a general tensor into its symmetric and antisymmetric parts.

Mixed Symmetry Types for Higher Order

For order three and above, the Schur-Weyl decomposition contains further pieces of mixed symmetry type, associated with partitions of d other than the two extremes; these mixed pieces are also separately invariant under the general linear group, but they play no role in the decomposition theory of purely symmetric tensors, since the projection onto the symmetric subspace discards them entirely. Their existence, however, clarifies that Subspace Invariance is not a special property unique to symmetry, but an instance of a much more general phenomenon governing every isotypic component of tensor space.

Implications for Symmetric Tensor Theory

Dimension Count as a Corollary

Because the symmetric subspace is a well-defined, invariant piece of tensor space, it has a well-defined dimension, independent of basis, computable as the number of degree-d monomials in n variables, given by the binomial coefficient of n plus d minus one, choose d; this dimension count, used throughout the theory to compare the space of symmetric tensors against the dimension of secant varieties of the Veronese variety in the Alexander-Hirschowitz classification, presupposes exactly the Subspace Invariance established here.

Foundation for the Symmetrization Projector

The existence of a well-defined symmetrization operator, projecting an arbitrary tensor onto the symmetric subspace by averaging over the permutation action, relies on the symmetric subspace being a genuine, invariant subspace rather than a coordinate-dependent artifact; Subspace Invariance guarantees that this projector, once composed with any change of basis, continues to land in the same intrinsically defined symmetric subspace, which is the structural fact silently used whenever the symmetric part of a tensor product is extracted in the construction of higher-order symmetric tensors discussed under the Symmetric Tensor Role of a quadratic form.