Antisymmetry

Antisymmetry refers to a property where swapping certain paired elements yields a collapse to equality or a sign change, depending on the mathematical setting. In mathematics, a binary relation R on a set S is antisymmetric if for all a,b in S, whenever aRb and bRa then a=b. This forbids distinct elements from standing in both directions. Antisymmetry is a central feature of partial orders, which also require reflexivity and transitivity; a total order is a partial order with every pair comparable.

Examples include the usual ≤ relation on real numbers, the divisibility relation | on the positive integers, and the subset relation ⊆ on sets. A relation that is symmetric aRb implies bRa is not antisymmetric unless a=b in all cases.

In linear algebra the term antisymmetric is often used in the sense of skew-symmetry for matrices: a square matrix A is antisymmetric (skew-symmetric) if A^T = -A. Such matrices have zero diagonal entries and eigenvalues that are purely imaginary or zero; they arise in areas such as Lie algebras and the study of oriented areas and cross products.

Antisymmetry also appears in tensor calculus: a tensor is antisymmetric in a set of indices if swapping those indices multiplies the tensor by -1. Antisymmetric tensors form the exterior algebra and underpin many geometric and physical constructions. In physics, antisymmetric wavefunctions describe fermions, reflecting the Pauli exclusion principle.