the first two spaces of the tensor product.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 and the second by Braiding To any tensor product on a single vector space, there are associated braiding maps. For example, the braiding map interchanges the two tensor factors (so that its action on simple tensors is given by ). 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 to denote the braiding map associated to the permutation (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 denote the Riemann tensor, regarded as a tensor in . The first Bianchi identity then asserts that 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 the Bianchi identity becomes Antisymmetrization and symmetrization 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 , where is the symmetric group on three elements. Similarly, we may symmetrize: See also Penrose graphical notation Einstein notation Index notation Tensor Antisymmetric tensor Raising and lowering indices Covariance and contravariance of vectors References Tensors Mathematical notation