16.20.4 Tensor Antisymmetrization Bracket Notation

Tensor antisymmetrization bracket notation is a concise way to express antisymmetric tensor components using permutation symbols and indices.

Tensor Antisymmetrization Bracket Notation is the specific symbolic device of enclosing a group of tensor indices in square brackets to denote the operation of summing over every permutation of those indices, each weighted by its permutation sign and normalized by the factorial of the number of indices involved, producing the fully antisymmetric part of an otherwise general tensor. It is the precise operator notation underlying the informal description of antisymmetrization used throughout alternating tensor theory.

Formal Definition of the Bracket

General n-Index Formula

For a tensor T with n indices enclosed in brackets, the antisymmetrization bracket is defined as:

T_{[i_{1} \dots i_{n}]} = \frac{1}{n!} \sum_{σ \in S_{n}} sgn (σ) T_{i_{σ (1)} \dots i_{σ (n)}}

The sum runs over all n! permutations of the n bracketed indices, and the normalizing factor 1/n! ensures the operation behaves as a projection rather than merely a scaled sum.

Two-Index Case

For the common case of two indices, the formula reduces to the simple antisymmetric combination:

T_{[i j]} = \frac{1}{2} (T_{i j} - T_{j i})

which directly matches the familiar formula for extracting the antisymmetric part of any two-index array.

Key Algebraic Properties

Idempotency

Applying the antisymmetrization bracket to an already antisymmetric tensor leaves it unchanged, since every permutation term in the defining sum already agrees, up to the correct sign, with the original tensor, and averaging identical values reproduces the same value:

{(T_{[i_{1} \dots i_{n}]})}_{[i_{1} \dots i_{n}]} = T_{[i_{1} \dots i_{n}]}

This idempotency confirms that the bracket operator behaves as a genuine projection onto the alternating subspace, rather than an operation that would compound or distort the result when applied repeatedly.

Linearity

The antisymmetrization bracket is linear: applying it to a sum of two tensors produces the sum of the individually antisymmetrized tensors, and applying it after scaling by a constant produces the same result as scaling the antisymmetrized tensor by that constant, consistent with the fact that the bracket is defined as a linear combination of permuted components.

Partial Bracket Notation

Excluding Indices From Antisymmetrization

When only a subset of a tensor's indices should be antisymmetrized while others remain fixed, vertical bars are used to exclude the fixed indices from the bracket, as in T_{i|k|j}, meaning antisymmetrization is performed over i and j while k is held out of the permutation sum entirely. This partial notation is essential for tensors combining mixed symmetry types across different groups of indices.

Nested and Multiple Brackets

More elaborate expressions may involve multiple separate bracket groups within a single tensor expression, each corresponding to antisymmetrization over a distinct subset of indices, allowing complex mixed-symmetry tensors, such as those arising in the study of Young tableaux and representation theory, to be expressed using combined bracket and parenthesis notation for their antisymmetric and symmetric index groups respectively.

Contrast With Symmetrization Notation

Parentheses for Symmetric Projection

Where square brackets denote antisymmetrization, parentheses conventionally denote symmetrization, defined analogously but without the sign factor sgn(σ), summing all permutations with equal positive weight. Comparing the two notations side by side highlights that antisymmetrization and symmetrization are complementary projection operations, both built from the same underlying permutation sum structure but differing in their sign weighting.

Significance of the Notation

Antisymmetrization bracket notation provides the exact, compact operator symbol for converting an arbitrary tensor into its antisymmetric component, encapsulating the full permutation sum and sign-weighting definition of alternation into a single bracketed index group. Its idempotency and linearity properties make it a genuine projection operator, its partial and nested forms accommodate mixed-symmetry tensors, and its contrast with symmetrization parentheses highlights the complementary structure underlying the decomposition of general tensors into symmetric and antisymmetric parts.