$\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers and Split Toric Varieties

For a smooth, projective scheme $X$ over a field $k$ or any variety $X$ if $k$ has characteristic zero, we compute the compactly supported $\mathbb{A}^1$-Euler characteristic of $\operatorname{Sym}^2(X)$ if $\operatorname{char}(k) \ne 2$ and of $\operatorname{Sym}^3(X)$ if $\operatorname{char}(k) \ne 2,3$. We do so by extending the definition of a $G$-equivariant quadratic Euler characteristic first studied by Pajwani-P\'al to arbitrary characteristic and by studying its relation to the $\mathbb{A}^1$-Euler characteristic of quotients. As an application, we show that the compactly supported $\mathbb{A}^1$-Euler characteristic of $\operatorname{Sym}^n(X)$ agrees with the prediction from the power structure constructed by Pajwani-P\'al for $n = 2,3$. Furthermore, we compute the compactly supported $\mathbb{A}^1$-Euler characteristic of split toric varieties and show that the compactly supported $\mathbb{A}^1$-Euler characteristic of all of their symmetric powers agrees with the prediction from the power structure constructed by Pajwani-P\'al.