16.6.5 Tensor Sign Change Alternating Preservation
Tensor Sign Change Alternating Preservation describes how tensor properties invert signs under transformations, key in algebraic symmetry and structure.
Tensor Sign Change Alternating Preservation is the fact that every operation compatible with the sign-change swap behavior produces a result that is itself alternating, meaning the property of being alternating is not lost or diluted when such tensors are combined, transformed, or manipulated through the standard operations of multilinear algebra.
Preservation Under Linear Combination
Sums and Scalar Multiples Remain Alternating
If S and T are both alternating tensors of the same rank k, any linear combination aS + bT also satisfies the swap behavior, since swapping arguments distributes linearly across the sum:
This confirms that the set of rank-k alternating tensors, Λᵏ(V), is closed under vector space operations and is therefore itself a vector space, not merely a collection of isolated examples.
Preservation Under the Wedge Product
Closure of Degree-Combined Products
The wedge product of two alternating tensors α (degree p) and β (degree q) is itself alternating, of degree p + q, precisely because the antisymmetrization built into the wedge product's definition enforces swap behavior on the combined argument list by construction:
Preservation Illustrated Through Graded Commutativity
The graded commutativity law α ∧ β = (−1)^{pq} β ∧ α is itself a manifestation of preserved swap behavior extended to blocks of arguments rather than single arguments; the sign factor (−1)^{pq} arises from counting the total number of individual argument transpositions needed to move all p arguments of α past all q arguments of β.
Preservation Under Linear Maps
Pullback Preserves Alternation
If f: W → V is a linear map and T is an alternating tensor on V, the pullback f*T, defined by precomposing with f in every argument, remains alternating on W:
since swapping two w arguments swaps the corresponding f(w) arguments fed into T, and T's own swap behavior takes over from there without any additional assumption on f.
Preservation Under Determinant-Scaled Pushforward
When f is an invertible linear map between spaces of the same dimension n, the top-degree alternating tensor transforms by an overall scalar equal to det(f), and this scaling preserves alternation exactly because scalar multiplication (already shown above) preserves the property.
Preservation Under Restriction and Antisymmetrization
Restricting to a Subspace
Restricting an alternating tensor's domain from V to a subspace U ⊆ V (evaluating only on vectors from U) preserves swap behavior trivially, since the restriction changes only which vectors are permitted as input, not the rule governing the tensor's response to argument swaps.
The Antisymmetrization Operator as a Preservation Guarantee
Applying the antisymmetrization operator Alt to any general (not necessarily alternating) tensor S produces a result that is guaranteed alternating regardless of whether S itself was:
Alt is, in effect, the universal preservation-enforcing operator: no matter the swap behavior of its input, its output always satisfies the alternating swap law.
What Preservation Excludes
Non-Preserving Operations
Not every operation preserves alternation: the ordinary (non-antisymmetrized) tensor product of two alternating tensors is generally not itself alternating, since it retains separate symmetry behavior in each factor's argument block without any cross-block antisymmetrization; this is precisely why the wedge product, not the plain tensor product, is used to combine alternating tensors while preserving the property.