15.2.6 Tensor Symmetric Notation Area

Tensor Symmetric Notation Area explains how symmetric tensors are compactly represented using algebraic notation to encode multilinear relationships.

Tensor Symmetric Notation Area is the collection of notational conventions used to write, manipulate, and abbreviate symmetric tensors so that their permutation invariance is visible directly in the symbols rather than left implicit. It covers index notation with symmetrization brackets, abstract index conventions, and compressed multi-index schemes such as Voigt notation, all of which exist to make symmetric structure explicit and to avoid redundant bookkeeping of permutation-equivalent components.

Symmetrization Brackets

Round Bracket Convention

The most direct notation for symmetrizing a set of indices uses round parentheses placed around the indices to be symmetrized. For a rank-2 tensor with components ( T_{ij} ), the symmetrized form is written:

T_{(i j)} = \frac{1}{2} (T_{i j} + T_{j i})

The parentheses signal that the enclosed indices are averaged over their own permutations, and this is the standard way to extract the symmetric part of an otherwise general tensor.

General Rank Symmetrization

For a rank-( r ) tensor, the notation extends to averaging over all ( r! ) permutations of the enclosed indices:

T_{(i_{1} i_{2} \dots i_{r})} = \frac{1}{r!} \sum_{σ \in S_{r}} T_{i_{σ (1)} \dots i_{σ (r)}}

where ( S_r ) denotes the symmetric group on ( r ) elements, and ( \sigma ) ranges over every permutation of the index positions.

Partial Symmetrization

When only some of the indices are meant to be symmetrized while others remain fixed, the fixed indices are set off from the symmetrization brackets, typically with vertical bars. An index excluded from symmetrization between two bracketed indices is written as:

T_{(i | k | j)}

indicating that ( i ) and ( j ) are symmetrized while ( k ) is held fixed and excluded from the permutation average.

Multi-Index Compression

Multi-Index Definition

Because a symmetric tensor's independent components are indexed by unordered selections rather than ordered tuples, notation frequently collapses a full ordered index list into a single multi-index ( \alpha = (\alpha_1, \ldots, \alpha_n) ), where each ( \alpha_k ) counts how many times the basis direction ( k ) appears among the ( r ) indices. This satisfies:

α_{1} + α_{2} + \dots + α_{n} = r

and a symmetric tensor is then written compactly as ( T_{\alpha} ) rather than repeating the fully expanded ordered-index form.

Voigt Notation

A widely used compressed notation for symmetric rank-2 tensors replaces the paired index ( (i,j) ) with a single running index, mapping each unordered pair to one label. For a symmetric tensor in three dimensions this gives the correspondence:

\begin{matrix} 1 \leftrightarrow (1, 1) \\ 2 \leftrightarrow (2, 2) \\ 3 \leftrightarrow (3, 3) \\ 4 \leftrightarrow (2, 3) \\ 5 \leftrightarrow (1, 3) \\ 6 \leftrightarrow (1, 2) \end{matrix}

reducing the nine components of a general rank-2 tensor to the six independent labels required by symmetry, and turning a two-index object into a single-index vector-like list.

Abstract and Diagrammatic Notation

Abstract Index Notation

In abstract index notation, symmetric tensors are written with lowercase Latin letters that label slots rather than numerical coordinate values, and symmetry is declared directly through the bracket convention rather than through any coordinate computation, so ( T^{(ab)} ) denotes the symmetric part of ( T^{ab} ) independent of any chosen basis.

Diagrammatic Symmetry Marking

Some notational systems mark symmetric index groups visually, connecting the symmetrized slots with a bracket or arc placed above or below the index string, signaling to the reader that any permutation among the connected slots leaves the component unchanged.

Notational Consistency Rules

Order Independence Requirement

Any valid symmetric notation must guarantee that reordering the labels inside the symmetrization bracket produces the identical symbol, since the underlying object is invariant under such reordering; notation that fails this requirement does not correctly represent a symmetric tensor.

Compatibility with Rank Area Counting

The multi-index and Voigt-style compressions are constructed precisely so that the number of distinct compressed labels equals the dimension of the symmetric rank area for the corresponding rank and dimension, ensuring the notation neither omits nor duplicates any independent component.