{
  "abstract": "We define a family of algebras $\\mathbb{Q}[\\mathbf{x}_n]/I_{n,a,b}$ whose Hilbert series, when $n\\le a+b\\le n+2$, $a>0$, and $0<b<n$, enumerate certain positroid cells by codimension. Equivalently, these series enumerate certain affine permutations by length. We determine the graded $\\mathfrak{S}_n$-module structure of these algebras by using exact sequences to derive an invertible recursion for their graded Frobenius characteristics when $a+b\\ge n$ and then passing to the limit $a\\rightarrow\\infty$. In particular, the graded Frobenius characteristic $\\mathrm{grFrob}(\\mathbb{Q}[\\mathbf{x}_{a+b-1}]/I_{a+b-1,a,b};q)=\\operatorname{rev}_q(ω((Θ_{e_{a-1}}Θ_{e_{b-1}}\\nabla e_1)\\mid_{t=0}))$, where $\\operatorname{rev}_q$ reverses the $q$-coefficients and the specialization $(Θ_{e_{a-1}}Θ_{e_{b-1}}\\nabla e_1)\\mid_{t=0}$ appears in the Theta conjecture. For $a,b\\ge1$, we also show that $e_a^\\perp\\mathrm{grFrob}(\\mathbb{Q}[\\mathbf{x}_{a+b}]/I_{a+b,a,b};q)=q^{\\binom{a}{2}}\\operatorname{rev}_q(ω((Δ_{e_{a-1}}^\\prime e_b)\\mid_{t=0}))$, where $Δ_{e_{k-1}}^\\prime e_n$ occurs in the Delta conjecture. In the three cases $a+b-n=0,1,2$, the ungraded Frobenius characteristics agree with three families of Eulerian quasisymmetric functions, respectively. We also construct an open subvariety of $(\\mathbb{P}^{k-1})^n$ whose rational Chow ring is isomorphic to $\\mathbb{Q}[\\mathbf{x}_n]/I_{n,k,n-k}$. This variety admits a stratification induced by certain positroids. Its strata need not be affine, and their codimension generating function equals $\\mathrm{Hilb}(\\mathbb{Q}[\\mathbf{x}_n]/I_{n,k,n-k};q)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21894",
  "author_total": 1,
  "authors": [
    {
      "name": "Hai Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3kc3prymebed9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21894",
  "fulltext_url": null,
  "id": "3kc3prymebed9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21894",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222658512",
  "pdf_url": "https://arxiv.org/pdf/2610.21894",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3kc3prymebed9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21894",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21894"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A family of algebras that connects positroids, the Theta conjecture, the Delta conjecture, and Eulerian quasisymmetric functions",
  "title_text": "A family of algebras that connects positroids, the Theta conjecture, the Delta conjecture, and Eulerian quasisymmetric functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/3kc3prymebed9",
  "venue": "arXiv",
  "versions": []
}