16.20.3 Tensor Exterior Power Notation

Tensor Exterior Power Notation is the collection of symbolic conventions used to denote exterior powers of a vector space and their elements, including the Λᵏ(V) space symbol, the wedge product operator, dual space and induced map notations, and the graded direct sum symbol used to describe the full exterior algebra. It provides the standardized written vocabulary for referring to exterior powers as spaces, as opposed to the index-based notation used for individual tensor components.

The Exterior Power Space Symbol

Basic Notation for a Single Degree

The k-th exterior power of a vector space V is denoted Λᵏ(V), with the capital Greek letter lambda representing the exterior algebra construction and the superscript k indicating the degree:

This notation is used consistently regardless of whether k is zero, giving the base field, one, giving V itself, or the ambient dimension n, giving the one-dimensional top exterior power.

The Full Graded Algebra Symbol

The entire exterior algebra, collecting every degree from 0 to n, is denoted simply Λ(V) without a superscript, or explicitly as a direct sum:

with the direct sum symbol ⊕ making explicit that Λ(V) is composed of independent graded pieces rather than a single homogeneous space.

Element and Operation Notation

Simple and General Elements

A simple, or decomposable, element of Λᵏ(V) formed by wedging k individual vectors is written v₁ ∧ v₂ ∧ ... ∧ vₖ, while a general element, not necessarily decomposable into a single wedge product, is typically denoted by a Greek letter such as ω, α, or β, reflecting its role as an arbitrary member of the vector space Λᵏ(V) rather than a specific wedge of named vectors.

Degree of an Element

When it is useful to indicate the degree of a general element explicitly, a subscript or superscript is attached, as in ωₖ or ω^(k), signaling that ω belongs specifically to the homogeneous component Λᵏ(V) of the graded algebra.

Dual Space and Functional Notation

Dual of the Exterior Power

The dual space of Λᵏ(V), consisting of linear functionals on k-vectors, is written (Λᵏ(V))*, and this space is naturally identified with the space of alternating k-linear forms, sometimes denoted Altᵏ(V) as an alternative notation emphasizing the multilinear function perspective rather than the dual vector space perspective.

Notation for the Determinant and Volume Forms

Within this notational system, the determinant and general volume forms are denoted as elements of (Λⁿ(V))*, often written simply as det or ω, with the understanding from context that they act as linear functionals on the one-dimensional top exterior power.

Induced Map Notation

Exterior Power of a Linear Map

Given a linear map T from V to W, the induced map on k-th exterior powers is denoted Λᵏ(T), mapping Λᵏ(V) to Λᵏ(W), and satisfying Λᵏ(T)(v₁ ∧ ... ∧ vₖ) = Tv₁ ∧ ... ∧ Tvₖ on simple elements:

Pullback Notation

The pullback of a form ω through a map T is written T*ω, with the asterisk superscript signaling that the operation moves in the opposite direction to T itself, from forms on the target space back to forms on the source space, a notational convention consistent across differential geometry and multilinear algebra.

Dimension and Basis Notation

Dimension Symbol

The dimension of an exterior power is written using the standard dimension symbol applied to the exterior power space, dim(Λᵏ(V)), and is understood to equal the binomial coefficient C(n, k) whenever V has finite dimension n.

Basis Multi-Index Notation

The alternating basis of Λᵏ(V), built from strictly increasing multi-indices, is denoted using the eᴵ notation described in alternating basis and multi-index conventions, tying the space-level notation of exterior powers directly to the component-level notation used for their explicit basis elements.

Significance of the Notation

Exterior power notation supplies the standardized symbolic vocabulary for referring to exterior powers as whole vector spaces, their elements, their duals, and the maps induced upon them by linear transformations. It complements the index-based alternating tensor notation used for individual components, together forming a complete symbolic toolkit that allows exterior algebra to be discussed and manipulated fluently at both the level of entire spaces and the level of specific numerical components.