HEXAGONbeta

On a non-basis of the coinvariant algebra

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

Contributed by Darij Grinberg ↗

SubmitterDarij Grinberg

Version 1 / Sep 29, 2026 / CC0 1.0

Abstract

A conjecture arising from a question of Procesi proposes a basis of the coinvariant algebra of SnS_n consisting of column-antisymmetrized monomials indexed by pairs of standard Young tableaux. We show that the proposed family need not even span an SnS_n-subrepresentation: for n=8n=8, a generator of degree 1515 is sent outside the span by the adjacent transposition (4,5)(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=7n=7, where the basis assertion first fails, although the span is still invariant.

Provenance statement

Written by GPT-5.6 Sol and subsequently expanded and revised by GPT-6 Astra and by myself. All claims have been verified with my own independently-written SageMath code.

Tools used

OpenAI
ChatGPTVersion 5.6 Sol
OpenAI
ChatGPTVersion 6 Astra
SageMath
SageMathVersion 10.10.beta9

References

  1. Edward E. Allen, https://doi.org/10.1073/pnas.89.9.3980 A conjecture of Procesi and the straightening algorithm of Rota , Proc. Natl. Acad. Sci. USA 89 (1992), no. 9, pp. 3980--3984.DOI
  2. Edward E. Allen, https://doi.org/10.1006/aima.1993.1035 A Conjecture of Procesi and a New Basis for the Decomposition of the Graded Left Regular Representation of $S_n$ , Advances in Mathematics 100 (1993), no. 2, pp. 262--292.DOI
  3. Susumu Ariki, Tomohide Terasoma, Hiro-Fumi Yamada, https://doi.org/10.32917/hmj/1206127144 Higher Specht polynomials , Hiroshima Mathematical Journal 27 (1997), pp. 177--188.DOI
  4. Accompanying verification scripts and exact outputs in the Appendix (Section sec.anc ). The standalone non-invariance check is procesi-n8-nonstability-verification.py ; see README.txt for the supplementary checks and instructions.

Version history

  1. v1Initial depositCurrentSep 29, 2026