Polynomials over Symmetric Polynomials
Let $\mathbf{k}$ be a commutative ring, and let the symmetric group $\mathfrak{S}_{n}$ act on $P=\mathbf{k}\left[ x_{1} ,x_{2},\ldots,x_{n} \right] $ by permuting the variables. We prove four results that are known (at least in the case when $\mathbf{k}$ is a field) but not easily found in the literature. First, the coinvariant algebra (the quotient of $P$ by the ideal generated by the symmetric polynomials with constant term $0$) is a free $\mathbf{k}$-module of rank $n!$, with the residue classes of the Artin monomials as a basis. Second, $P$ is a free module of rank $n!$ over the ring…
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