Alternating polynomial

Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation:

In particular, alternating polynomials form a module over the algebra of symmetric polynomials (the odd part of a superalgebra is a module over the even part); in fact it is a free module of rankĀ 1, with the Vandermonde polynomial in n variables as generator.

If the characteristic of the coefficient ring is 2, there is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials.

This is clearly alternating, as switching two variables changes the sign of one term and does not change the others.[2]

Conversely, the ratio of two alternating polynomials is a symmetric function, possibly rational (not necessarily a polynomial), though the ratio of an alternating polynomial over the Vandermonde polynomial is a polynomial. Schur polynomials are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial.

That is, the ring of symmetric and alternating polynomials is a quadratic extension of the ring of symmetric polynomials, where one has adjoined a square root of the discriminant.

From the perspective of representation theory, the symmetric and alternating polynomials are subrepresentations of the action of the symmetric group on n letters on the polynomial ring in n variables. (Formally, the symmetric group acts on n letters, and thus acts on derived objects, particularly free objects on n letters, such as the ring of polynomials.)

The symmetric group has two 1-dimensional representations: the trivial representation and the sign representation. The symmetric polynomials are the trivial representation, and the alternating polynomials are the sign representation. Formally, the scalar span of any symmetric (resp., alternating) polynomial is a trivial (resp., sign) representation of the symmetric group, and multiplying the polynomials tensors the representations.

In characteristic 2, these are not distinct representations, and the analysis is more complicated.