# Alternating Polynomial - Relation To Symmetric Polynomials

Relation To Symmetric Polynomials

Products of symmetric and alternating polynomials (in the same variables ) behave thus:

• the product of two symmetric polynomials is symmetric,
• the product of a symmetric polynomial and an alternating polynomial is alternating, and
• the product of two alternating polynomials is symmetric.

This is exactly the addition table for parity, with "symmetric" corresponding to "even" and "alternating" corresponding to "odd". Thus, the direct sum of the spaces of symmetric and alternating polynomials forms a superalgebra (a -graded algebra), where the symmetric polynomials are the even part, and the alternating polynomials are the odd part. This grading is unrelated to the grading of polynomials by degree.

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 rank 1, with as generator the Vandermonde polynomial in n variables.

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.