16.11.2 Tensor Exterior Product Antisymmetrized Tensor Product
The antisymmetrized tensor product, also known as the exterior product, constructs alternating tensors by eliminating symmetric components through contraction.
Tensor Exterior Product Antisymmetrized Tensor Product is the explicit realization of the wedge product as the ordinary tensor product composed with antisymmetrization, α ∧ β = c · Alt(α ⊗ β) for an appropriate normalization constant c, making precise the sense in which the exterior product is "the tensor product, but forced to be alternating."
The Two Operations Being Compared
The Plain Tensor Product
Given α ∈ Λᵖ(V*) and β ∈ Λᵍ(V*), viewed simply as multilinear forms, their ordinary tensor product α ⊗ β is the rank-(p+q) multilinear form defined by:
This product is not alternating in general, even when both α and β are: it treats the first p arguments and the last q arguments as belonging to separate, non-interacting blocks, with no antisymmetry imposed across the block boundary.
The Wedge Product as Its Corrected Version
The exterior product recovers alternation by applying Alt to the plain tensor product, with a normalization constant restoring the correct scale:
Why the Plain Tensor Product Fails to Be Alternating
A Concrete Counterexample
Take α = φ and β = ψ, two covectors (degree 1 each), so α⊗β is a rank-2 tensor. Swapping its two arguments gives (φ⊗ψ)(v,u) = φ(v)ψ(u), which generally does not equal −φ(u)ψ(v) = −(φ⊗ψ)(u,v) unless φ and ψ satisfy a special relationship; in general φ⊗ψ has no fixed symmetry type at all.
Cross-Block Arguments Break Alternation
Even when α and β are each individually alternating, α⊗β fails to be alternating specifically because swapping one argument from α's block with one from β's block is not accounted for by either α's or β's individual alternating property; alternation within each block does not imply alternation across the block boundary.
How Antisymmetrization Repairs This
Averaging Over All Cross-Block Rearrangements
Applying Alt to α⊗β sums over every possible permutation of all p+q arguments, including every way of interleaving the first p slots with the last q slots; this interleaving is precisely what is missing from the plain tensor product, and summing over it with the correct signs is what restores full alternation across the combined argument list.
Normalization Ensures Consistency with Decomposable Formula
The binomial coefficient (p+q)!/(p!q!) in front of Alt(α⊗β) is chosen precisely so this formula matches the determinant-of-pairings evaluation formula for decomposable wedges exactly, rather than differing from it by some leftover scalar multiple.
Worked Example at Rank 2
Explicit Computation
For two covectors φ and ψ (each degree 1, so p=q=1), the formula gives:
directly recovering the familiar bivector formula φ∧ψ = φ⊗ψ − ψ⊗φ from the general antisymmetrized-tensor-product definition.