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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.

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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 $m,n\geq 0$.

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  • $\begingroup$ What is it meant by "divide"? $\endgroup$
    – Rubilax96
    Mar 27, 2023 at 21:00
  • $\begingroup$ @Rubilax96 A polynomial $P$ divides another polynomial $Q$ if we can write $Q=P\cdot R$ for some polynomial $R$. $\endgroup$ Mar 27, 2023 at 23:10

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