Almost symmetric polynomial?

Question

Let's say we have a polynomial $(x-y)(y-z)(x-z)$. This is not a symmetric polynomial, but it almost is. Every permutation of the variables results in a polynomial whose factors are multiples of the factors of the original polynomial. For example, if we send $y$ to $x$ and $x$ to $y$, we get $(y-x)(x-z)(y-z)=-1(x-y)(y-z)(x-z)$. In fact, each factor is multiplied by $1$ or $-1$. Is there a name for this kind of polynomial?

Answer 1

It's called an alternating polynomial, or sometimes a skew-symmetric polynomial.

Whenever a polynomial reacts to every permutation of the variables by multiplication of with some constant factor, it must be either symmetric or alternating. This is because the constant factors for each permutation must constitute a one-dimensional representations of $S_n$, and the only such representations are the trivial representation and the sign representation.