{
  "abstract": "We study quotients of free associative algebras by the two-sided ideals generated by positive-degree symmetric, quasisymmetric, or free quasisymmetric polynomials. Free quasisymmetric functions give a word realization of the Malvenuto-Reutenauer Hopf algebra of permutations. For quotients on $n$ variables, finiteness in the symmetric and quasisymmetric cases follows from a theorem of Bergman--Isaacs when $n!$ is invertible in the field. Our main result is a nilpotence criterion for families of relations compatible with restriction to subalphabets. The criterion proves finiteness over every field for quasisymmetric, $r$-quasisymmetric, and free quasisymmetric coinvariants, with explicit degree bounds exponential in $n$ for fixed $r$. In the quasisymmetric and $r$-quasisymmetric cases, these improve the bounds inherited from Bergman--Isaacs. A principal application is to the free quasisymmetric case, where, for $n\\geq3$, the group-invariant argument used for the symmetric case does not apply. For the symmetric quotient $C_n$, we prove that invertibility of $n!$ is also necessary for finiteness. In characteristic zero, an auxiliary presentation and the Golod--Shafarevich inequality give $\\log\\dim C_n\\geq(1-o(1))n^5(\\log n)^2$ as $n\\to\\infty$. The same inequality shows that quasisymmetric and free quasisymmetric coinvariant dimensions grow faster than every fixed power of $n!$. For all three families on several freely noncommuting alphabets, we determine all multigraded Hilbert series and, for symmetric coinvariants in characteristic zero, Frobenius characteristics from one-alphabet data. In characteristic zero, projection gives stable Schur multiplicities for compatible quotients in mixed commuting, anticommuting, and free alphabets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16873",
  "author_total": 4,
  "authors": [
    {
      "name": "Farid Aliniaeifard"
    },
    {
      "name": "Nantel Bergeron"
    },
    {
      "name": "Huilan Li"
    },
    {
      "name": "Shu Xiao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/39px6qcnwqf7n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16873",
  "fulltext_url": null,
  "id": "39px6qcnwqf7n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16873",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221803409",
  "pdf_url": "https://arxiv.org/pdf/2610.16873",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/39px6qcnwqf7n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:16Z",
      "native_id": "2610.16873",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16873"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Finite noncommutative coinvariant algebras",
  "title_text": "Finite noncommutative coinvariant algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:16Z",
  "url": "https://research.draconitic.com/paper/39px6qcnwqf7n",
  "venue": "arXiv",
  "versions": []
}