16.13 Tensor Exterior Power Structure

Tensor Exterior Power Structure is the organization of Λᵏ(V), the k-th exterior power of a vector space, as a construction in its own right — its universal property among alternating multilinear maps, its behavior under linear maps between vector spaces, and its place within the broader family of tensor constructions alongside symmetric powers and ordinary tensor powers.

The Exterior Power as a Universal Object

The Universal Property

Λᵏ(V) is characterized by a universal property: for any vector space W, alternating multilinear maps V^k → W correspond exactly to linear maps Λᵏ(V) → W, via composition with the canonical alternating map V^k → Λᵏ(V) sending (v₁,...,vₖ) to v₁∧...∧vₖ:

This universal property is what makes Λᵏ(V) the "most general" or "freest" possible receptacle for alternating multilinear data: any alternating multilinear map factors uniquely through it.

Consequence: Uniqueness Up to Isomorphism

The universal property determines Λᵏ(V) uniquely up to a canonical isomorphism; any two constructions satisfying the same universal property (the quotient construction and the antisymmetrization-image construction, for instance) must be isomorphic in a way compatible with the canonical alternating map, explaining why different textbook definitions of the exterior power all agree.

Functorial Behavior Under Linear Maps

Induced Maps Between Exterior Powers

A linear map f: V → W induces a linear map Λᵏ(f): Λᵏ(V) → Λᵏ(W) on exterior powers, defined on decomposable elements by:

and extended linearly to all of Λᵏ(V).

Functoriality Confirmed

This assignment respects composition, Λᵏ(g∘f) = Λᵏ(g)∘Λᵏ(f), and sends the identity map to the identity map, confirming Λᵏ(−) is a genuine functor from vector spaces to vector spaces, not merely a construction applied case by case without any compatibility across different spaces.

Effect on Isomorphisms

If f is an isomorphism, Λᵏ(f) is also an isomorphism, since the induced map on the top exterior power Λⁿ(f) is given by scalar multiplication by det(f), and more generally Λᵏ(f) is invertible whenever f is, with inverse Λᵏ(f⁻¹).

Relationship to Sibling Constructions

Exterior Power versus Symmetric Power

The exterior power Λᵏ(V) sits alongside the symmetric power Symᵏ(V) as one of the two extreme quotients of the ordinary tensor power V^{⊗k}: Λᵏ(V) imposes full antisymmetry, Symᵏ(V) imposes full symmetry, and both are obtained from V^{⊗k} by quotienting by different ideals or projecting via different idempotent operators.

Dimension Comparison

with Λᵏ(V) always finite-dimensional and eventually zero as k exceeds n, whereas Symᵏ(V) grows without bound as k increases.

The Full Exterior Power Family

Assembling Every k Simultaneously

Taken together across all k from 0 to n, the exterior powers assemble into the graded exterior algebra Λ(V) = ⊕ₖ Λᵏ(V), with the wedge product supplying the multiplication connecting different powers; this structure is the natural home for every alternating tensor discussed throughout the broader theory, unifying the individual exterior powers into a single coherent algebraic object.