Let $n$ be a positive integer and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$. For any $w\in S_n$ and polynomial $f\in \mathbb{R}[x_1,x_2,\ldots,x_n]$, denote $w(f)=f(x_{w(1)},x_{w(2)},\ldots,x_{w(n)})$. Furthermore, define $$\alpha(f)=\sum\limits_{w\in S_n}\epsilon(w)w(f),$$ where the sum is over all the $w\in S_n$ and $\epsilon(w)$ denotes the sign of the permutation $w$.

It is easy to see that for any $f\in \mathbb{R}[x_1,x_2,\ldots,x_n]$, $\alpha(f)$ is divisible by $\prod\limits_{1\leq i<j\leq n}(x_i-x_j).$ So when $f$ is a homogeneous polynomial of degree $\frac{n(n-1)}{2}$ , we have $$\alpha(f)=a_f\prod\limits_{1\leq i<j\leq n}(x_i-x_j)$$ for some $a_f\in \mathbb{R}$ .

Suppose $f$ is a homogeneous polynomial of degree $\frac{n(n-1)}{2}$, how to know whether $a_f$ is $0$ or not when the expansion $$f(x_1,x_2,\ldots,x_n)=\sum\limits_{i_1+i_2+\cdots+i_n=\frac{n(n-1)}{2}}a_{i_1,i_2,\ldots,i_n}x_1^{i_1}x_2^{i_2}\cdots x_n^{i_n}$$ is not easy to get?

For example, let $n=5$ and $$f(x_1,x_2,x_3,x_4,x_5)=(x_1+x_2)^2(x_2+x_3)^2(x_3+x_4)^2(x_4+x_5)^2(x_5+x_1)^2,$$ then $f$ is homogeneous polynomial of degree $10$.

With the expansion of $f$ expanded by the computer, I can calculate that $a_f$ is nonzero. But how to get this without the expansion of $f$?

Maybe it is hard to answer my question in general, so I hope someone can answer my question for the specific polynomial I gave above.