DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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The wedge product of alternating covectors

Definition

If $α \in {Alt}^{k} (V)$ and $β \in {Alt}^{ℓ} (V)$ , their wedge product is

$α \land β : = \frac{(k + ℓ)!}{k! ℓ!} Alt (α \otimes β) .$

Equivalently,

$(α \land β) (v_{1}, \dots, v_{k + ℓ}) = \sum_{σ \in Sh (k, ℓ)} sgn (σ) α (v_{σ (1)}, \dots, v_{σ (k)}) β (v_{σ (k + 1)}, \dots, v_{σ (k + ℓ)}),$

where $Sh (k, ℓ)$ is the set of $(k, ℓ)$ -shuffles.

Depends on

Used by

The exterior algebra of covectors Definition
The wedge product of differential forms Definition
The wedge product is alternating and bilinear Lemma
Exterior-power duality pairing Proposition
Interior product is a graded antiderivation Proposition
The wedge product is associative and graded commutative Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)