# Antisymmetric polynomials in two variables

In Prove that every symmetric polynomial can be written in terms of the elementary symmetric polynomials it is argued that a symmetric polynomial of two variables $$x,y$$ can be written as a sum over terms of the form $$(x+y)^m (xy)^n$$, for $$m,n \ge0$$ integers.

Is there anything similar for antisymmetric polynomials in two variables? I would like to have one (or many equivalent) parametrization in terms of elementary antisymmetric "blocks" like $$x+y$$, $$xy$$ in the symmetric case. Of course, such parametrization should give the most general antisymmetric polynomial for a given order.

One such block could be given for example by $$(x-y)^m$$ for odd integers $$m$$, in turn multiplied by any $$x,y$$ symmetric term.

If $$f(x,y)$$ is an anti-symmetric polynomial, then the polynomial $$x-y$$ divides $$f$$, and $$f$$ can be written as $$(x-y)g(x,y)$$, where $$g$$ is a symmetric polynomial. (See here, for example, for a generalization to $$n$$ variables.) So, using the characterization of symmetric polynomials, any anti-symmetric polynomial can be written as a linear combination of polynomials of the form $$(x-y)(x+y)^m(xy)^n$$ for integers $$m,n\geq 0$$.
• @Rubilax96 A polynomial $P$ divides another polynomial $Q$ if we can write $Q=P\cdot R$ for some polynomial $R$. Mar 27, 2023 at 23:10