16.20.1 Tensor Alternating Index Notation

Tensor Alternating Index Notation is a concise way to express antisymmetric tensor properties using indexed variables and sign conventions.

Tensor Alternating Index Notation is the specific system of subscripted and superscripted indices used to write the components of alternating tensors, incorporating conventions for covariant and contravariant placement, antisymmetrization brackets, and implicit ordering rules that distinguish it from the index notation used for general, unrestricted tensors. It is the detailed indexing framework that makes it possible to write compact, unambiguous expressions for antisymmetric tensor components and the operations performed on them.

Basic Index Placement

Covariant and Contravariant Positions

Following standard tensor notation conventions, lower indices denote covariant components, transforming with the basis, while upper indices denote contravariant components, transforming with the dual basis. An alternating tensor of covariant rank k is written with k lower indices, such as Tᵢⱼₖ, while an alternating tensor of contravariant rank k, such as an element of Λᵏ(V) itself, is written with k upper indices, such as Tⁱʲᵏ.

Implied Antisymmetry Convention

When an index notation expression is declared or understood to represent an alternating tensor, the antisymmetry of the indices is implicit: writing Tᵢⱼ for an alternating 2-tensor already carries the understanding that Tᵢⱼ = −Tⱼᵢ, without needing to restate this relation explicitly every time the symbol appears in a longer expression.

Antisymmetrization Bracket Notation

Square Bracket Convention

Square brackets enclosing a set of indices denote antisymmetrization over those indices, meaning the alternating sum over all permutations of the enclosed indices divided by the factorial of the number of indices involved:

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

This notation allows an arbitrary tensor to be converted into its alternating projection using a compact bracket symbol, without spelling out the sum over permutations explicitly each time.

Partial Antisymmetrization

The square bracket convention can also be applied to a subset of indices within a larger expression, with indices excluded from antisymmetrization typically separated by vertical bars, as in T_{i|k|j}, indicating that antisymmetrization is applied to indices i and j while k is held fixed and excluded from the permutation sum. This partial bracket notation is especially useful when working with tensors that combine both symmetric and antisymmetric index structure.

Distinguishing From General Tensor Index Notation

Absence of Explicit Symmetry Assumptions

For a general, unrestricted tensor, index notation carries no implicit assumption about how the value changes under a permutation of indices; each component is treated as independent unless a symmetry or antisymmetry property is separately stated. Alternating index notation departs from this default by building the antisymmetry assumption directly into the meaning of the symbol whenever a tensor is declared to be alternating.

Reduced Independent Component Listing

Because alternating index notation carries the antisymmetry assumption implicitly, expressions and formulas involving alternating tensors are typically written referencing only strictly increasing index sequences, relying on the reader's understanding that all other index arrangements are determined by the sign and vanishing rules rather than needing separate specification.

Index Notation in Common Formulas

The Levi-Civita Symbol as Alternating Index Notation

The Levi-Civita symbol ε_{i₁...iₙ} is itself an example of pure alternating index notation, with its value determined entirely by the parity of the permutation formed by its indices, and it serves as the standard tool for converting between wedge product expressions and explicit indexed sums.

Wedge Product in Index Form

The components of a wedge product of two alternating tensors S and T, of ranks p and q respectively, are written in index notation using antisymmetrization brackets across the combined index set:

(S \land T)_{i_{1} \dots i_{p + q}} \propto S_{[i_{1} \dots i_{p}} T_{i_{p + 1} \dots i_{p + q}]}

with an overall proportionality constant depending on convention, expressing the exterior product entirely in terms of indexed components and bracket antisymmetrization.

Significance of the Notation

Alternating index notation, with its covariant and contravariant placement rules, bracket antisymmetrization convention, and implicit sign assumptions, provides the precise and efficient symbolic system needed to write, combine, and manipulate alternating tensor expressions without repeatedly restating their defining antisymmetry properties. It underlies standard formulas throughout differential geometry and theoretical physics, distinguishing itself from general tensor notation through its built-in assumption of antisymmetric index behavior.