{"id":"hexagon/2609.00121","pdf":"https://hexagonmath.org/pdf/2609.00121v1","url":"https://hexagonmath.org/abs/2609.00121v1","tags":["math.CO","msc:05E40","msc:20C30","msc:13P10","Artin basis","Gröbner bases","coinvariant algebra","orbit harmonics","symmetric polynomials"],"score":0,"title":"Polynomials over Symmetric Polynomials","domain":"math.AC","origin":"ai-generated","source":"hexagon","authors":["Darij Grinberg"],"reviews":0,"subject":"math.AC","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-09-28","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}