HEXAGONbeta

Polynomials over Symmetric Polynomials

A mix of human-written and AI-generated textHuman understanding: all partsmath.AC — Commutative Algebramath.CO — Combinatorics

Contributed by Darij Grinberg ↗

Version 1 / Sep 28, 2026 / CC0 1.0

Abstract

Let k\mathbf{k} be a commutative ring, and let the symmetric group Sn\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 k\mathbf{k} is a field) but not easily found in the literature. First, the coinvariant algebra (the quotient of PP by the ideal generated by the symmetric polynomials with constant term 00) is a free k\mathbf{k}-module of rank n!n!, with the residue classes of the Artin monomials as a basis. Second, PP is a free module of rank n!n! over the ring PSnP^{\mathfrak{S}_{n}} of symmetric polynomials, again with the Artin monomials as a basis. Third, if n!n! is invertible in k\mathbf{k}, the coinvariant algebra is the regular $\mathbf{k}\left[ \mathfrak{S}_{n} \right] $-module. Fourth, under the same hypothesis, PP is a free left PSn[Sn]P^{\mathfrak{S}_{n}}\left[ \mathfrak{S}_{n} \right] -module of rank 11. The first result follows from an elementary normal-form lemma for monic polynomials with pairwise relatively prime leading monomials. The second is proved by lifting the Artin basis. For the third, we use orbit harmonics with a strongly discrete point orbit, and the fourth follows by equivariantly lifting a regular basis of the coinvariant algebra.

Provenance statement

Generated by GPT-5.6 Sol, subsequently refined over a long conversation and proofread by myself. This is a semi-expository work; the only novelty is generalizing the known results to arbitrary commutative rings, which may well have been implicit in existing literature. This also appears on my website as https://www.cip.ifi.lmu.de/~grinberg/algebra/poloversym-gpt.pdf / https://www.cip.ifi.lmu.de/~grinberg/algebra/poloversym-gpt.tex .

Tools used

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ChatGPTVersion 5.6 Sol

References

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Version history

  1. v1Submitted by Darij GrinbergInitial depositCurrentSep 28, 2026