= Polynomial invariants of the standard S4 representation
{title2=$\mathbb C[W]^{S_4}=\mathbb C[p_2,p_3,p_4]$}
Realize the <standard representation of the symmetric group> on $W=\{x\in\mathbb C^4:\sum_ix_i=0\}$. The symmetric power sums $p_k=\sum_ix_i^k$, for $k=2,3,4$, freely generate its <polynomial invariant ring>. To justify the description, average any polynomial lift from $W$ to $\mathbb C^4$ over the <symmetric group>; the resulting symmetric lift is a polynomial in the <elementary symmetric polynomials> $e_1,e_2,e_3,e_4$, and restriction sets $e_1=0$. The remaining three generators are algebraically independent. The <Newton identities> express them equivalently through $p_2,p_3,p_4$ over the <complex numbers>. In coordinates $x=(a,b,c,-a-b-c)$, one has $p_2=a^2+b^2+c^2+(a+b+c)^2$ and $p_3=-3(a+b)(b+c)(c+a)$.
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