antisymmetrized

Antisymmetrized refers to an object that has been made antisymmetric with respect to exchange of certain indices by applying the antisymmetrization operation. For a tensor T with k indices, the fully antisymmetric part is denoted T_{[i1 i2 ... ik]} and is defined by

T_{[i1 i2 ... ik]} = 1/k! sum_{π in S_k} sgn(π) T_{i_{π(1)} i_{π(2)} ... i_{π(k)}},

where S_k is the permutation group and sgn(π) is the permutation sign. This construction guarantees that swapping any pair of indices multiplies the component by −1. For two indices, T_{[ij]} = (T_{ij} − T_{ji})/2, and T_{ii} = 0.

Antisymmetrization projects a tensor onto the space of alternating multilinear forms. The operation is a projection operator and underlies the exterior algebra, where antisymmetric tensors are combined via the wedge product. Examples include a ∧ b with components a_i b_j − a_j b_i for two vectors. In general, a totally antisymmetric k-tensor on a d-dimensional space has dimension C(d, k) and vanishes if any two indices are equal.

In practical terms, antisymmetrized objects appear in many areas. Differential geometry uses differential forms, which are completely antisymmetric tensors; in physics, antisymmetrized wavefunctions describe fermions and are implemented in constructions such as Slater determinants. The Levi-Civita symbol ε_{i1...id} provides a standard basis for fully antisymmetric tensors in d dimensions.