16.9 Tensor Alternating Multilinear Form Structure

Tensor Alternating Multilinear Form Structure is the general algebraic organization of alternating forms of arbitrary rank k, extending the bilinear (rank-2) case to any number of arguments and assembling the resulting family of spaces, operations, and invariants into the full exterior algebra framework.

The General Multilinear Form

Defining the k-Argument Map

An alternating multilinear form of rank k is a map T: V^k → 𝔽 that is linear in each of its k arguments and satisfies the alternating condition on every pair of them:

for every position of u and v among the k slots. The bilinear case is exactly k = 2, and this structure recovers it as the smallest genuinely nontrivial instance.

The Family Across All Ranks

For each k from 0 to n = dim(V), the alternating multilinear forms of rank k form a vector space Λᵏ(V*), and assembling all ranks together gives the graded structure:

Dimension and Component Structure

Dimension Count at Each Rank

Each layer of the multilinear form structure has dimension given by the binomial coefficient:

generalizing the rank-2 count C(n,2) established for bilinear forms to every rank simultaneously.

Basis via Ordered Index Tuples

A basis for Λᵏ(V*) is given by the wedge products e*_{i₁} ∧ ... ∧ e*_{iₖ} for increasing index tuples i₁ < ... < iₖ, directly generalizing the eᵢ* ∧ eⱼ* basis used for bilinear forms.

The Wedge Product as the Structural Multiplication

Combining Forms of Different Rank

The wedge product extends the multilinear form structure into a full associative algebra, taking a rank-p form and a rank-q form to a rank-(p+q) form:

restricting to the ordinary bilinear-form wedge product exactly when p = q = 1, showing bilinear forms as the product of two rank-1 covectors.

Evaluation Formula at General Rank

The Determinant-of-Minors Formula

For decomposable forms built as a wedge of covectors φ₁ ∧ ... ∧ φₖ, evaluation on k vectors reduces to a determinant of a k × k matrix of pairings:

generalizing the two-dimensional area formula for bilinear forms to signed k-volumes at arbitrary rank.

Structural Bookends of the Family

Rank 0 and Rank 1

At k = 0, the structure reduces to scalars, with no alternation condition to check. At k = 1, it reduces to ordinary covectors V*, again with no alternation condition, since a single argument has no partner to swap against.

Rank n: The Top Structural Layer

At k = n, the structure collapses to a one-dimensional space, and every nonzero element serves as a volume form on V, generalizing the nondegenerate case seen at rank 2 (symplectic structure) to the unique top-degree case at rank n.

Diagram of the Full Multilinear Form Family