antisymmetrization

Antisymmetrization is a mathematical operation that extracts the antisymmetric, or alternating, part of a tensor. It is commonly defined on the n-fold tensor product V ⊗ V ⊗ ... ⊗ V of a vector space V over a field, using the antisymmetrization operator A = (1/n!) sum over all permutations π of {1,...,n} of sgn(π) P_π, where P_π permutes the tensor factors and sgn(π) is the sign of the permutation. The operator A projects onto the space Λ^n V of alternating tensors, also known as the nth exterior power. This space consists of tensors that change sign under any swap of two arguments and vanish if two inputs are identical in a simple tensor form.

For a simple tensor v1 ⊗ v2 ⊗ ... ⊗ vn, antisymmetrization yields A(v1 ⊗ ... ⊗ vn) = (1/n!) sum_{π} sgn(π) v_{π(1)} ⊗ ... ⊗ v_{π(n)}. An important consequence is that antisymmetrized tensors vanish when any two arguments coincide, reflecting the alternating property. For n = 2 this reduces to v ⊗ w − w ⊗ v, which generates the wedge product v ∧ w in Λ^2 V.

In differential geometry and algebra, Λ^n V forms the exterior algebra, where the wedge product provides a natural multiplication. In physics, antisymmetrization underpins the construction of fermionic wavefunctions and Slater determinants, in accordance with the Pauli exclusion principle.

Key properties include A^2 = A, and the dimension of Λ^n V is the binomial coefficient C(dim V, n). Antisymmetrization thus serves to isolate the antisymmetric components of tensors and plays a central role in many mathematical and physical theories.