Alternating Polynomial - Vandermonde Polynomial

Vandermonde Polynomial

The basic alternating polynomial is the Vandermonde polynomial:

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

The alternating polynomials are exactly the Vandermonde polynomial times a symmetric polynomial: where is symmetric. This is because:

• is a factor of every alternating polynomial: is a factor of every alternating polynomial, as if, the polynomial is zero (since switching them does not change the polynomial, we get

so is a factor), and thus is a factor.

• an alternating polynomial times a symmetric polynomial is an alternating polynomial; thus all multiples of are alternating polynomials

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.