4.20.3 Tensor Alternating Sign Change

Tensor Alternating Sign Change describes tensors whose sign flips when indices are swapped, a key feature in multilinear algebra and differential geometry.

Tensor Alternating Sign Change is the phenomenon by which an alternating multilinear map's output is multiplied by the sign of a permutation, +1 or -1, whenever its arguments are reordered according to that permutation. Sign change is the precise quantitative content of alternation once attention moves from single transpositions to arbitrary reorderings, and it is governed entirely by the sign homomorphism from the symmetric group to {+1, -1}.

The Sign Homomorphism

Definition via Transpositions

Every permutation σ of {1, ..., n} can be written as a product of transpositions, and although this decomposition is not unique, the parity of the number of transpositions used is always the same for a given σ. This parity defines the sign

sgn (σ) = {(- 1)}^{t}

where σ is any product of t transpositions, and this value does not depend on which decomposition is used.

Homomorphism Property

The sign function satisfies sgn(στ) = sgn(σ)sgn(τ), making it a group homomorphism from the symmetric group Sₙ to the multiplicative group {+1, -1}; a permutation with sgn(σ) = +1 is called even, and one with sgn(σ) = -1 is called odd, and the even permutations form a subgroup, the alternating group Aₙ, of index two in Sₙ.

Sign Change for Alternating Multilinear Maps

The General Rule

For an alternating multilinear map f: V × ... × V → W and any permutation σ ∈ Sₙ,

f (v_{σ (1)}, \dots, v_{σ (n)}) = sgn (σ) f (v_{1}, \dots, v_{n})

The output of an alternating map under any reordering of its inputs is determined completely by the original output together with a single sign, computed abstractly from the permutation alone, independent of the map f and the vectors vᵢ.

Cycle Type Determines Sign Directly

The sign of a permutation can be computed from its cycle decomposition without explicitly listing transpositions: a cycle of length k has sign (-1)^{k-1}, and the sign of σ is the product of the signs of its disjoint cycles. This gives a fast route to the sign change a given reordering induces on an alternating map, bypassing the need to count transpositions directly.

Sign Change and Orientation

Volume and Orientation Reversal

For the determinant, viewed as the alternating n-linear map sending n vectors in an n-dimensional space to the signed volume of the parallelepiped they span, a sign change under permutation corresponds exactly to a reversal of orientation: an even permutation of the spanning vectors preserves the geometric orientation of the parallelepiped, while an odd permutation reverses it, with the magnitude of the volume unaffected either way.

Sign Change Under a Change of Basis

Applying a linear map A: V → V to all n arguments of an alternating n-linear map on an n-dimensional space scales the output by det(A):

f (A v_{1}, \dots, A v_{n}) = det (A) f (v_{1}, \dots, v_{n})

so a linear map with negative determinant induces exactly the same kind of sign change on the top alternating form that an odd permutation of its arguments would, connecting sign change under permutation to sign change under orientation-reversing linear transformations of the whole space.

Sign Change as a Representation of the Symmetric Group

The Sign Representation

The rule σ · c = sgn(σ) c for c ∈ F defines the one-dimensional sign representation of Sₙ; the top exterior power ⋀ⁿV of an n-dimensional space, on which Sₙ acts by permuting a chosen basis with the accompanying sign, is exactly this representation, one-dimensional and spanned by any single wedge product e₁ ∧ ... ∧ eₙ of a basis.

Contrast With the Trivial Representation

Symmetric multilinear maps correspond instead to the trivial representation, σ · c = c for every σ, under which no sign change occurs regardless of the permutation applied. The distinction between the sign representation and the trivial representation is exactly the distinction between alternating and symmetric multilinear maps, with general multilinear maps of arity n ≥ 3 decomposing into a richer collection of irreducible representations of Sₙ beyond just these two.

Practical Computation of Sign Change

Counting Inversions

For a permutation given explicitly as a rearrangement of 1, ..., n, its sign can be computed as (-1)^{inv(σ)}, where inv(σ) counts the number of inversions, pairs i < j with σ(i) > σ(j); this method gives a direct algorithm for determining the sign change an alternating map undergoes under any explicitly specified reordering of its arguments.

Tracking Sign During Row Reduction

When computing a determinant via row reduction, each row swap performed contributes a factor of -1 to the running sign, accumulated across all swaps used to reach an upper triangular form; this bookkeeping is a direct, algorithmic application of alternating sign change, converting the abstract permutation-sign rule into a concrete step tracked during a numerical computation.

Field Characteristic Considerations

Characteristic Two Collapses the Distinction

Since -1 = 1 in characteristic 2, the sign sgn(σ) ∈ {+1, -1} becomes trivial, +1 regardless of σ, and sign change ceases to distinguish even from odd permutations at the level of scalar multiplication; alternating maps are still defined via vanishing on repeated arguments in this setting, but the sign-change formulation of the same condition loses its distinguishing power and the vanishing-on-repeats definition becomes the operative one.