It was remarked in Section 5.1 that the determinant function is the classic example of a multilinear function. In addition to its multilinearity, the determinant function has another characteristic feature—it is alternating. This chapter is devoted to an examination of tensors that have a corresponding property.

7.1 Multicovectors

Let V be a vector space of dimension m, and let s ≥ 1 be an integer. Following Section B.2, we denote by the group of permutations on {1, … , s}. For each permutation σ in , consider the linear map

defined by

(7.1.1)

for all tensors in and all vectors v ₁, ..., v _s in V. By saying that σ is linear, we mean that for all tensors , ℬ in and all real numbers c,

There is potential confusion arising from (7.1.1), as a simple example illustrates. Let s = 3, let be a tensor in , and consider the permutation σ = (1 2 3) in . According to (7.1.1),

for all vectors v ₁, v ₂, v ₃ in V. To be consistent, it seems that should be interpreted as , but this is incorrect. The issue is that the indices in (v ₁, v ₃, v ₂) are not sequential, which is implicit in the way (7.1.1) is presented. Setting (w ₁, w ₂, w ₃) = (v ₁, v ₃, v ₂), we have from (7.1.1) that

In most of the computations that follow, the indices are sequential, thereby avoiding this issue.

We say that a tensor in is symmetric if for all permutations σ in ; that is,

for all vectors v ₁, … , v _s in V. We denote the set of symmetric tensors in by ∑ _s (V). It is easily shown that ∑ _s (V) is subspace of .

A tensor ℬ in is said to be alternating if τ(ℬ) =  − ℬ for all transpositions τ in , or equivalently, if

for all vectors v ₁,…, v _s in V for all 1 ≤ i < j ≤ s. The set of alternating tensors in is denoted by Λ ^s (V). It is readily demonstrated that Λ ^s (V) is a subspace of . An element of Λ ^s (V) is called an s‐covector or multicovector. When s = 1, the alternating criterion is vacuous, and a 1‐covector is simply a covector. Thus,

(7.1.2)

Since Λ ^s (V) is a subspace of , the zero multicovector in Λ ^s (V) is precisely the zero tensor in . A multicovector in Λ ^s (V) is nonzero when it is nonzero as a tensor in . To be consistent with (5.1.2), we define

(7.1.3)

With these definitions, the determinant function det : (Mat _m × 1) ^m  → ℝ is seen to be a multicovector in Λ ^m (Mat _{m  ×  1}).

There are several equivalent ways to characterize multicovectors inΛ ^s (V).

We now introduce a way of associating a multicovector to a given tensor. Let V be a vector space. Alternating map is the family of linear maps

defined for s ≥ 0 by

(7.1.4)

for all tensors in .

In view of Theorem 7.1.3(b), we can replace the map in (7.1.4) with

7.2 Wedge Products

In Section 5.1, we introduced a type of multiplication of tensors called the tensor product. Our next task is to define a corresponding operation for multicovectors.

Let V be a vector space. Wedge product is the family of linear maps

defined for s, s′ ≥ 0 by

(7.2.1)

for all multicovectors η in Λ ^s (V) and ζ in . That is,

for all vectors v ₁,..., v _s+s′ in V, where the first equality follows from (7.2.1), and the second equality from (5.1.4) and (7.1.1).

Wedge products behave well with respect to basic algebraic structure.

Any operation that purports to be a type of “multiplication” should be associative. The wedge product meets this requirement.

In light of the associativity of the wedge product, we drop parentheses and, for example, denote (η ∧ ζ) ∧ ξ and η ∧ (ζ ∧ ξ) by η ∧ ζ ∧ ξ , with corresponding notation for wedge products of more than three terms.

The next result is a generalization of Theorem 7.2.5.

The next result shows that wedge products and determinants are closely related, which is not so surprising.

Part (a) of the next result is a generalization of Theorem 1.2.1(e).

7.3 Pullback of Multicovectors

The pullback of covariant tensors was briefly considered in Section 5.2. The corresponding theory for multicovectors is far richer.

Before proceeding, we pause to consider multi‐index notation, which was discussed in Section 2.1 in the context of matrices. Let V be a vector space, let (h ₁,…,h _m ) be a basis for V, and let (θ ¹,…,θ ^m ) be its dual basis. For an integer 1 ≤ s ≤ m, let I = (i ₁,…,i _s ) be a multi‐index in ℐ _{s, m} , and let us denote

and

In this notation, the unordered basis for Λ ^s (V) in Theorem 7.2.12(a) can be expressed concisely as {θ ^I  : I ∈ ℐ _{s, m} }, and the identity in Theorem 7.2.13(a) becomes

If we order ℐ _{s, m} in some fashion, for example, lexicographically, then the basis {θ ^I  : I ∈ ℐ _{s, m} } of Theorem 7.2.12(a) can be similarly ordered, yielding an ordered basis for Λ ^s (V), which we denote by

Kronecker's delta can be generalized to the multi‐index setting. Let J = (j ₁,…,j _s ) be another multi‐index in ℐ _{s, m} , and define

Then the conditional identity in Theorem 7.2.9 is simply . As discussed in Appendix A, the complement of the multi‐index (i) in ℐ_{1, m} is

for i = 1,…, m. In this notation, the multicovectors comprising the basis in Theorem 7.2.12(e) can be expressed as . We will find multi‐index notation of great utility in what follows.

Let V and W be vector spaces, and let A : V → W be a linear map. Pullback by A (for multicovectors) is the family of linear maps

defined for s ≥ 1 by

(7.3.1)

for all multicovectors η in Λ ^s (W) and all vectors v ₁,…,v _s in V. It follows from the multilinearity and alternating properties of η and the linearity of A that A*(η) is in Λ ^s (V), so the definition makes sense. We refer to A*(η) as the pullback of η by A .

Special cases of Theorem 7.3.2 provide a number of useful identities.

The next result is reminiscent of Theorem 2.4.9.

7.4 Interior Multiplication

Let V be a vector space of dimension m, and let v be a vector in V. Interior multiplication by v is the family of linear maps

defined for s ≥ 2 by

for all multicovectors η in Λ ^s (V) and all vectors w ₁,…,w _s–1 in V. Since any m + 1 vectors in V are linearly dependent, it follows from Theorem 7.3.5 that i _v is the zero map when s > m. Recalling from (7.1.2) and (7.1.3) that Λ¹(V) = V* and Λ⁰(V) = ℝ, we extend the preceding definition to s = 1 as follows:

is given by

for all covectors η in Λ¹(V). For s = 0, we trivially define i _v = 0. Let us denote

Not surprisingly, interior multiplication behaves well with respect to basic algebraic structure, but its handling of wedge products is more complex.

The next result shows that interior multiplication satisfies a novel product rule.

7.5 Multicovector Scalar Product Spaces

In Section 4.5, we showed how to construct the dual of a scalar product space. Building on that foundation, we now generalize to multicovectors.

Let (V, ) be a scalar product space, let ℋ be a basis for V, and let Θ = (θ ¹,…,θ ^m ) be its dual basis. By Theorem 7.2.12(a), (θ ^I : I ∈ ℐ _{s, m} ) is a basis for Λ ^s (V). We define a bilinear function

as follows. For multicovectors η, ζ in Λ ^s (V), with

(7.5.1)

let

(7.5.2)

where * is the scalar product on V* and

is the matrix of * with respect to Θ.

It would appear that the definition of ^Λ is dependent on the choice of basis for V. Remarkably, this turns out not to be the case:

We are now in a position to define the multicovector counterpart of the dual scalar product space described in Section 4.5.

Now that we have the scalar product space (Λ ^s (V), g ^Λ), the construction in Section 4.5 yields the corresponding dual scalar product space (Λ ^s (V)^*, g ^Λ*) and the associated flat map and sharp map:

By definition,

(7.5.12)

for all multicovectors η, ζ in Λ ^s (V).