Hexagon

math.CO — Combinatorics

On a non-basis of the coinvariant algebra

Contributed by Darij Grinberg

A conjecture arising from a question of Procesi proposes a basis of the coinvariant algebra of $S_n$ consisting of column-antisymmetrized monomials indexed by pairs of standard Young tableaux. We show that the proposed family need not even span an $S_n$ -subrepresentation: for $n=8$ , a generator of degree $15$ is sent outside the span by the adjacent transposition $(4,5)$ . The proof is an exact finite computation with an explicit separating functional, requiring neither a rank computation nor Gr\"obner reduction. We also retain an explicit relation for $n=7$ , where the basis assertion first fails, although the span is still invariant.

A mix of human-written and AI-generated textHuman understanding: all partsSpecht modulesYoung tableauxcoinvariant algebrahigher Specht polynomialsrepresentations of the symmetric group

Sep 29, 2026PDF ↗View work →

hexagon:2609.00121v1

math.AC — Commutative Algebra

Polynomials over Symmetric Polynomials

Contributed by Darij Grinberg

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 $P^{\mathfrak{S}_{n}}$ of symmetric polynomials, again with the Artin monomials as a basis. Third, if $n!$ is invertible in $\mathbf{k}$ , the coinvariant algebra is the regular $\mathbf{k}\left[ \mathfrak{S}_{n} \right] $-module. Fourth, under the same hypothesis, $P$ is a free left $P^{\mathfrak{S}_{n}}\left[ \mathfrak{S}_{n} \right]$ -module of rank $1$ . 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.

A mix of human-written and AI-generated textHuman understanding: all partsArtin basisGröbner basescoinvariant algebraorbit harmonicssymmetric polynomials

Sep 28, 2026PDF ↗View work →

Matching works

On a non-basis of the coinvariant algebra

Polynomials over Symmetric Polynomials

Advanced search

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory