{"id":"hexagon/2609.00137","pdf":"https://hexagonmath.org/pdf/2609.00137v1","url":"https://hexagonmath.org/abs/2609.00137v1","tags":["math.AC","msc:20C30","msc:05E40","Specht modules","Young tableaux","coinvariant algebra","higher Specht polynomials","representations of the symmetric group"],"score":0,"title":"On a non-basis of the coinvariant algebra","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Darij Grinberg"],"reviews":0,"subject":"math.CO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","ChatGPT","SageMath"],"discussed":0,"published":"2026-09-29","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}