The identity can be considered a a product of no transpositions, or the product of P ij with its inverse P ij , or the product of any number of transpositions paired with their inverses. It is not hard to show that any permuation can be expressed as a product of transpositions and although the factors in the product are not unique the number of them is either even or odd. In this way we assign a parity to every permuation. Some important properties of permutations and the antisymmetrizer are listed here without proof. The parity of P and that of P -1 are the same.
Any permutation commutes with operators that are symmetric in system coordinates.
Hence also the antisymmetrizer commutes with such operators. The antisymmetrizer operator does not just apply to orbital products. It can be used to project the fermion symmetry from any trial function. If the trial function is a Hartree product then the projection is a Slater determinant. But if the trial function is, say, a function that explicitly contains the distance between pairs of electrons, then the projection cannot be expressed as a single Slater determinant.