15.19.4 Tensor Symmetrization Bracket Notation

Tensor Symmetrization Bracket Notation is the round-parenthesis convention placed around a set of tensor indices to denote that those indices have been averaged over all their permutations, together with its variants for symmetrizing over only a subset of indices while excluding others, and the differing normalization conventions found across the literature.

The Basic Bracket

Full Symmetrization of All Indices

Enclosing all d indices of a tensor's component in round parentheses denotes the fully symmetrized combination introduced in the general Tensor Symmetric Tensor Notation:

If T is already symmetric, this bracketed expression simply equals T itself, since every term of the average already coincides; the bracket notation becomes meaningful precisely when applied to a tensor that is not already known to be symmetric, in which case it produces the projection of that tensor onto the symmetric subspace, consistent with Subspace Invariance.

Bracket Applied to a General, Possibly Non-Symmetric Tensor

Written this way, the bracket notation supplies a compact symbol for the symmetrization operator itself, allowing an expression such as T subscript with bracketed indices to appear inside a larger formula without needing to separately name and define the symmetrization map each time it is used, a convenience exploited repeatedly whenever a general tensor product must be projected onto S^d V in the course of building higher-order symmetric tensors, as discussed under the Symmetric Tensor Role of a quadratic form.

Partial Symmetrization

Excluding Indices with Vertical Bars

When only a subset of a tensor's indices should be symmetrized, leaving the remaining indices fixed in place, the excluded indices are set off with vertical bars within the bracket:

denotes symmetrization over the first and third index positions only, while the second index position, enclosed by vertical bars, is held fixed and excluded from the averaging; this partial notation is essential whenever a tensor carries indices of genuinely different roles, such as a tensor built by combining a symmetric block with an additional, unrelated index, and only the symmetric block's indices should participate in the symmetrization.

Symmetrization Over a Specified Sub-Block

More generally, the bracket can enclose any chosen subset of the full index list, with all indices outside the bracket, whether or not set off by vertical bars, understood to be held fixed throughout the averaging; this generalized partial bracket is the notation used when constructing tensors of mixed symmetry type, as classified by the partitions discussed under Tensor Symmetric Type Preservation, in which some groups of indices are symmetrized among themselves while remaining independent of other groups.

Normalization Conventions

The Normalized Convention

The convention presented above, dividing by d factorial (or, for a partial symmetrization over k indices, by k factorial), produces a true projection operator, idempotent in the sense that applying it twice gives the same result as applying it once, and this normalized convention is standard whenever the bracket is meant to represent an actual orthogonal projection onto a symmetric subspace, as required for consistency with the basis normalization discussed under the Symmetric Basis Notation.

The Unnormalized Convention

Some sources, particularly in classical tensor calculus and general relativity, omit the dividing factorial altogether, using the bracket to denote simply the unweighted sum over permutations; under this convention, the bracketed expression is d factorial times the normalized version, and care must be taken, whenever consulting or combining material from different sources, to establish which convention is in force, since formulas that are correct under one convention will carry a spurious combinatorial factor if reinterpreted under the other.

Contrast with Antisymmetrization Notation

Square Brackets for the Opposite Extreme

The bracket notation for symmetrization is paired with an analogous square-bracket notation for antisymmetrization, in which the permutation sum is instead weighted by the sign of each permutation, producing the projection onto the antisymmetric subspace discussed as the complementary piece under Tensor Symmetric Subspace Invariance; the visual pairing of round and square brackets throughout the literature is a deliberate mnemonic, with round brackets suggesting the unsigned, fully symmetric average and square brackets suggesting the signed, alternating average.

Mixed Bracket Expressions

Expressions combining both bracket types, symmetrizing over one group of indices while antisymmetrizing over another, arise when constructing tensors of the mixed symmetry types associated with partitions other than the two extremes, and such mixed expressions rely on both bracket conventions being fixed and unambiguous, reinforcing the importance of stating the adopted normalization convention explicitly whenever the Symmetrization Bracket Notation is used in a formal or comparative context.