16.10.2 Tensor Antisymmetrization Sign Weight

Tensor Antisymmetrization Sign Weight is the sgn(σ) factor attached to each term of the antisymmetrization permutation sum, isolated here as a standalone weighting function that determines whether a given reordering of arguments contributes positively or negatively to the final antisymmetrized result.

The Weight Function Itself

Definition as a Map on Permutations

The sign weight is a function from the symmetric group S_k to {+1, −1}:

assigning +1 to even permutations and −1 to odd permutations, with no other values possible.

Weight as a Group Homomorphism

The sign weight is not an arbitrary assignment; it is a group homomorphism, meaning it respects composition of permutations:

This homomorphism property is what guarantees the weights assigned to different terms in the antisymmetrization sum compose consistently as permutations are built up from transpositions.

Role of the Weight in the Sum

Selecting the Contribution of Each Term

In the permutation sum defining Alt(S), the sign weight determines the coefficient multiplying each rearranged copy of S:

Half of the k! terms (the even permutations) enter the sum with weight +1; the other half (the odd permutations) enter with weight −1.

Weight Determines Cancellation Patterns

Whether a particular contribution to the sum survives or cancels against another depends entirely on the weight assigned to each permutation: two terms with opposite weights but otherwise identical values cancel exactly, which is precisely the mechanism by which Alt annihilates fully symmetric tensors.

Computing the Weight Directly

Counting Inversions

For a permutation given explicitly as a sequence, the sign weight can be computed by counting inversions — pairs of positions (i,j) with i < j but σ(i) > σ(j):

giving a direct computational recipe for the weight without needing to explicitly decompose σ into transpositions first.

Weight via Cycle Structure

Alternatively, the weight can be computed from the cycle decomposition of σ: a cycle of length ℓ contributes weight (−1)^{ℓ−1}, and the total weight of σ is the product of the weights of its disjoint cycles.

The Weight's Kernel: The Alternating Group

Even Permutations Form a Subgroup

The set of permutations with weight +1 — the kernel of the sign weight homomorphism — forms a subgroup of S_k called the alternating group A_k, of index 2. This subgroup is exactly the set of rearrangements that leave any alternating tensor's value unchanged.

Coset Structure

The odd permutations form the single remaining coset of A_k in S_k; every odd permutation can be written as a fixed transposition composed with some even permutation, reflecting the two-valued nature of the weight function.

Why the Weight Must Be Exactly Sign, Not Any Other Function

Uniqueness Among One-Dimensional Representations

The sign weight is (together with the trivial weight, always +1) one of only two one-dimensional representations of S_k for k ≥ 2; no other homomorphism from S_k to a multiplicative group of scalars exists, which is why antisymmetrization has exactly one natural sign-weighting scheme available to it, rather than a family of possible alternatives.