 More generally, given a function in "n" variables, one can symmetrize by taking the sum over all formula_13 permutations of the variables, or anti-symmetrize by taking the sum over all formula_14 even permutations and subtracting the sum over all formula_14 odd permutations (except that when , the only permutation is even).Here symmetrizing a symmetric function multiplies by formula_13 – thus if formula_13 is invertible, such as when working over a field of characteristic formula_18 or formula_19, then these yield projections when divided by formula_13.In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for formula_21 there are others – see representation theory of the symmetric group and symmetric polynomials.