16.9.1 Tensor Alternating Multilinear Argument Set

Tensor Alternating Multilinear Argument Set is the generalization of the bilinear argument pair to an ordered k-tuple of vectors (v₁, ..., vₖ) fed into a rank-k alternating form, extending the earlier pairwise analysis of ordering, dependence, and geometric meaning to argument sets of arbitrary size.

The Argument Set as an Ordered k-Tuple

Ordering Matters at Every Rank

Just as an argument pair (u,v) is distinct from (v,u), an argument set (v₁,...,vₖ) for a rank-k alternating form is an ordered tuple, and any two distinct orderings of the same underlying k vectors are, in general, different inputs to the form:

The alternating property links the values of T across different orderings of the same set of k vectors by a sign, but treats each ordering as a genuinely distinct point of V^k until that linkage is applied.

The Sign Orbit of a Fixed Underlying Set

For a fixed unordered set of k linearly independent vectors, there are k! distinct orderings, and the alternating property partitions these k! orderings into two groups of equal size: those related to a reference ordering by an even permutation (sharing its value) and those related by an odd permutation (taking the negated value).

Classifying Argument Sets by Dependence

Independent Argument Sets

An argument set is called independent if v₁,...,vₖ are linearly independent; for such sets, the alternating form may take any value consistent with its definition, and the set spans a genuine k-dimensional subspace of V.

Dependent Argument Sets

If the k vectors satisfy any linear relation Σcᵢvᵢ = 0 with not all cᵢ zero, bilinearity combined with the vanishing rule forces the alternating form to evaluate to zero on the entire set, generalizing the dependent-pair case from bilinear forms to any rank:

Oversized Argument Sets

If k exceeds n = dim(V), every argument set of size k is automatically dependent, so an alternating form of rank greater than n evaluates to zero on literally every argument set available to it, consistent with the general rank ceiling established for alternating tensors.

Geometric Reading of the Argument Set

The Set as a Spanned Parallelepiped

An independent argument set (v₁,...,vₖ) naturally represents the k-dimensional parallelepiped spanned by these vectors, generalizing the parallelogram picture from the bilinear case; the value of a top-rank alternating form (the determinant) on this set gives the signed k-volume of that parallelepiped.

Reordering as Reflected Traversal

Just as swapping a pair reverses traversal direction for a parallelogram, applying an odd permutation to a k-argument set corresponds to a reflection of the associated parallelepiped's orientation, while an even permutation preserves it — the geometric meaning of the sign law extends unchanged from rank 2 to general rank k.

Contraction and Restriction of Argument Sets

Fixing Some Arguments, Varying Others

An argument set can be partially fixed, with some of the k slots held constant and the remainder left as free variables; the resulting object is again alternating in the free slots, since the sign-reversal law applies independently to any two of them regardless of what values the fixed slots hold.

Subsets and the Restriction to Fewer Arguments

Selecting a sub-tuple of j < k vectors from a larger argument set does not, by itself, produce a value from the rank-k form directly; instead, this operation corresponds to a different construction (interior product or contraction), which uses the removed vectors to produce a rank-(k−j) form rather than simply restricting the original form's argument set.

Diagram of Argument Set Classification