Let $W=\mathfrak{S}_n$ be the symmetric group of $n$-elements. Can we construct the coinvariant ring for it? Say, $$\mathbb{Q}[x_1,\ldots,x_n]\bigg/(e_1,\ldots,e_n)$$ where $V=\mathbb{C}^n$ is the ambient space and the denominator stands for the ideal generated by the positive degree part of invariant polynomials.
Moreover, it is well-known that the coinvariant ring is spanned by monomials $x^{\lambda}=x_1^{\lambda_1}x_2^{\lambda_2}\cdots $ with $\lambda_i\leq \rho_i=n+1-i$.
How to construct an algorithm to reduce any polynomial to the space spanned by them?
We definitely can do so by Schubert polynomials, but it is far from effective. I tried to search in Sage, GAP, and Macaulay 2, but found nothing.