16.6.3 Tensor Sign Change Even Permutation Rule
The Tensor Sign Change Even Permutation Rule describes how tensor components transform under even permutations, preserving their sign and reflecting algebraic structure.
Tensor Sign Change Even Permutation Rule is the complementary case of the general sign-change law, applying when the rearrangement of an alternating tensor's arguments is an even permutation — one requiring an even number of transpositions to construct — in which case the rule guarantees the tensor's output value is left completely unchanged from its value on the original argument order.
Defining Even Permutation Input
What Makes a Permutation Even
A permutation σ of k objects is classified as even if any (equivalently, every) decomposition of σ into transpositions uses an even number of them:
The identity permutation, corresponding to m = 0, is itself an even permutation by this counting convention, since zero is an even number.
The Rule Itself
Statement of the Even Permutation Rule
For an alternating tensor T of rank k and any even permutation σ of its k arguments:
No matter how elaborate the rearrangement, provided the underlying permutation is even, the tensor's value is exactly reproduced — not merely up to some other constant, but identically.
Derivation from Transposition Composition
Because σ decomposes into an even number m of transpositions, and each transposition contributes a factor of −1 under the sign-change behavior:
the product of an even number of −1 factors returns to +1.
Recognizing Even Permutations in Practice
Double Transposition
The simplest nontrivial even permutation, beyond the identity, is a pair of disjoint transpositions applied together (m = 2), leaving the tensor's value fully unchanged despite two separate slot exchanges having occurred.
Odd-Length Cycles Are Even Permutations
A cycle of odd length k decomposes into k − 1 transpositions, an even count; consequently, any 3-cycle, 5-cycle, or other odd-length cycle is classified as an even permutation, meaning it leaves an alternating tensor's value unchanged despite visibly rearranging three or more arguments at once:
illustrating the 3-cycle case, where the sign remains +1 even though every one of the three arguments has moved to a new position.
Contrast with the Odd Permutation Case
The Complementary Rule
Even permutations are paired with the complementary case of odd permutations, for which the sign-change rule instead negates the tensor's value entirely.
Composition of Two Even Permutations Gives Even
Composing two even permutations always yields another even permutation, since their transposition counts sum to an even total; this closure property is exactly what makes the set of even permutations form the alternating subgroup A_k within the full symmetric group S_k.
The Alternating Group Connection
The name "alternating group" for A_k, the subgroup of even permutations, is historically and structurally linked to alternating tensors precisely because it is this subgroup that leaves an alternating tensor's value unchanged; the sign homomorphism sgn: S_k → {+1, −1} has A_k exactly as its kernel.