Abstract index notation

A general homogeneous tensor is an element of a tensor product of copies of ${\displaystyle V}$ and ${\displaystyle V^{*}}$, such as

${\displaystyle V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.}$

Label each factor in this tensor product with a Latin letter in a raised position for each contravariant ${\displaystyle V}$ factor, and in a lowered position for each covariant ${\displaystyle V^{*}}$ position. In this way, write the product as

${\displaystyle V^{a}V_{b}V_{c}V^{d}V_{e}}$

or, simply

${\displaystyle V^{a}{}_{bc}{}^{d}{}_{e}.}$

The last two expressions denote the same object as the first. Tensors of this type are denoted using similar notation, for example:

${\displaystyle h^{a}{}_{bc}{}^{d}{}_{e}\in V^{a}{}_{bc}{}^{d}{}_{e}=V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.}$

In general, whenever one contravariant and one covariant factor occur in a tensor product of spaces, there is an associated contraction (or trace) map. For instance,

${\displaystyle \mathrm {Tr} _{12}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}\otimes V\otimes V^{*}}$

is the trace on the first two spaces of the tensor product.

$${\displaystyle \mathrm {Tr} _{15}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}\otimes V^{*}\otimes V}$$

is the trace on the first and last space.

These trace operations are signified on tensors by the repetition of an index. Thus the first trace map is given by

${\displaystyle \mathrm {Tr} _{12}:h{}^{a}{}_{b}{}_{c}{}^{d}{}_{e}\mapsto h{}^{a}{}_{a}{}_{c}{}^{d}{}_{e}}$

and the second by

${\displaystyle \mathrm {Tr} _{15}:h{}^{a}{}_{b}{}_{c}{}^{d}{}_{e}\mapsto h{}^{a}{}_{b}{}_{c}{}^{d}{}_{a}.}$

To any tensor product on a single vector space, there are associated braiding maps. For example, the braiding map

${\displaystyle \tau _{(12)}:V\otimes V\rightarrow V\otimes V}$

interchanges the two tensor factors (so that its action on simple tensors is given by ${\displaystyle \tau _{(12)}(v\otimes w)=w\otimes v}$). In general, the braiding maps are in one-to-one correspondence with elements of the symmetric group, acting by permuting the tensor factors. Here, we use ${\displaystyle \tau _{\sigma }}$ to denote the braiding map associated to the permutation ${\displaystyle \sigma }$ (represented as a product of disjoint cyclic permutations).

Braiding maps are important in differential geometry, for instance, in order to express the Bianchi identity. Here let ${\displaystyle R}$ denote the Riemann tensor, regarded as a tensor in ${\displaystyle V^{*}\otimes V^{*}\otimes V^{*}\otimes V}$. The first Bianchi identity then asserts that

${\displaystyle R+\tau _{(123)}R+\tau _{(132)}R=0.}$

Abstract index notation handles braiding as follows. On a particular tensor product, an ordering of the abstract indices is fixed (usually this is a lexicographic ordering). The braid is then represented in notation by permuting the labels of the indices. Thus, for instance, with the Riemann tensor

${\displaystyle R=R_{abc}{}^{d}\in V_{abc}{}^{d}=V^{*}\otimes V^{*}\otimes V^{*}\otimes V,}$

the Bianchi identity becomes

${\displaystyle R_{abc}{}^{d}+R_{cab}{}^{d}+R_{bca}{}^{d}=0.}$

A general tensor may be antisymmetrized or symmetrized, and there is according notation.

We demonstrate the notation by example. Let's antisymmetrize the type-(0,3) tensor ${\displaystyle \omega _{abc}}$, where ${\displaystyle \mathrm {S} _{3}}$ is the symmetric group on three elements.

${\displaystyle \omega _{[abc]}:={\frac {1}{3!}}\sum _{\sigma \in \mathrm {S} _{3}}(-1)^{{\text{sgn}}(\sigma )}\omega _{\sigma (a)\sigma (b)\sigma (c)}}$

Similarly, we may symmetrize:

${\displaystyle \omega _{(abc)}:={\frac {1}{3!}}\sum _{\sigma \in \mathrm {S} _{3}}\omega _{\sigma (a)\sigma (b)\sigma (c)}}$

• Penrose graphical notation
• Einstein notation
• Index notation
• Tensor
• Antisymmetric tensor
• Raising and lowering indices
• Covariance and contravariance of vectors