{
  "abstract": "We prove Bergeron's Hook-Components and Coefficient-Length conjectures for the Vandermonde polarization module and full diagonal harmonics in every rank and number of alphabets. Hook-label characters are elementary skewings of the alternating character, and the maximal Schur length in label $μ$ is $n-μ_1$. We compute character and dimension polynomials of diagonal coinvariants in every bosonic--fermionic superspace for $D\\ge3$, $n\\ge\\max\\{D,2D-6\\}$, answering Church--Ellenberg--Farb's character and dimension questions. For augmented commutative algebras in characteristic zero, we construct natural two-Schur presentations and filtered free resolutions with one Schur generator in each homological degree. For polynomial algebras, specialization to orthogonal idempotents identifies the first unstable correction for every nonempty tail $τ$ and degree $D>|τ|$. The stable threshold $n\\ge D+τ_1$ is sharp when the row dimension is at least $\\ell(τ)$. First parabolic algebras recover finite jets, their multiplication and their maps. Selected-label lifts give uniform Vandermonde generation through degree $D$ for $D\\ge3$, $n\\ge2D-3$, and disjoint-label products generate every first occurrence for positive integral row weights. Constructive degree reflection determines all two-column moment homology and generates its first syzygies from degree $n+1$; two-column Vandermonde generation reduces to degree $n-1$. Invariants and harmonics separate through the character range and through degree $n$ in two alphabets. The hook identities also give the fully graded sign component of the $t=0$, $(1,2)$-superspace Theta conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00629",
  "author_total": 1,
  "authors": [
    {
      "name": "Eren Ercan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a138gaqh2hxhw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00629",
  "fulltext_url": null,
  "id": "a138gaqh2hxhw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00629",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00629",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a138gaqh2hxhw.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00629",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00629"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Multivariate diagonal harmonics: character polynomials and proofs of two conjectures of Bergeron",
  "title_text": "Multivariate diagonal harmonics: character polynomials and proofs of two conjectures of Bergeron",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/a138gaqh2hxhw",
  "venue": "arXiv",
  "versions": []
}