16.6.2 Tensor Sign Change Odd Permutation Rule
The Tensor Sign Change Odd Permutation Rule describes how tensor components change sign under odd permutations, a fundamental property in multilinear algebra.
Tensor Sign Change Odd Permutation Rule is the specific case of the general sign-change law that applies when the rearrangement of an alternating tensor's arguments is an odd permutation — one requiring an odd number of transpositions to construct — in which case the rule guarantees the tensor's output value is negated relative to its value on the original argument order.
Defining Odd Permutation Input
What Makes a Permutation Odd
A permutation σ of k objects is classified as odd if any (equivalently, every) decomposition of σ into transpositions uses an odd number of them:
The parity of m is an invariant of σ itself, not of the particular decomposition chosen, which is what allows "odd permutation" to be a well-defined classification.
The Rule Itself
Statement of the Odd Permutation Rule
For an alternating tensor T of rank k and any odd permutation σ of its k arguments:
Every odd permutation, regardless of how many elements it actually moves or how it is structured as a product of cycles, produces exactly this single sign flip — no more, no less.
Derivation from Transposition Composition
Since σ decomposes into an odd number m of transpositions, and the sign-change behavior applied to each transposition contributes a factor of −1:
the product of an odd number of −1 factors is itself −1, giving the rule directly.
Recognizing Odd Permutations in Practice
Single Transposition
The most basic odd permutation is a single transposition itself, m = 1, immediately odd, giving the familiar two-argument swap rule as the smallest instance of this general rule.
Even-Length Cycles Are Odd Permutations
A cycle of even length k decomposes into k − 1 transpositions, an odd count; consequently, any 4-cycle, 6-cycle, or other even-length cycle is classified as an odd permutation and falls under this rule, even though intuitively a "cycle" might seem like a single unified operation rather than a composite of swaps.
illustrating the 4-cycle case, where the sign is −1 despite no single transposition appearing explicitly in the written rearrangement.
Contrast with the Even Permutation Case
The Complementary Rule
Odd permutations are paired with the complementary case of even permutations, for which the sign-change rule instead preserves the original value unchanged:
Every permutation is either odd or even, never both and never neither, so together the two rules exhaustively determine the sign-change output for any rearrangement whatsoever.
Composition of Two Odd Permutations Gives Even
A structural consequence worth noting: composing two odd permutations always yields an even permutation, since their transposition counts sum to an even total, matching the general sign homomorphism property sgn(σ∘τ) = sgn(σ)·sgn(τ).