{
  "abstract": "We compute the homological Euler characteristics of $\\mathrm{SL}_4(\\mathbb{Z})$ and $\\mathrm{GL}_4(\\mathbb{Z})$ with coefficients in arbitrary irreducible rational highest-weight representations. Applying Wall's formula, we combine the orbifold Euler characteristics of centralizers of torsion elements with traces computed using the Jacobi-Trudi identity to derive explicit formulas and rational generating functions. Consequently, these Euler characteristics are quasi-polynomial functions of the highest-weight parameters, of total degree at most two. We further derive degreewise vanishing results and parity-sensitive lower bounds for the dimensions of cohomology groups. The two extensions of an $\\mathrm{SL}_4(\\mathbb{Z})$-coefficient system to $\\mathrm{GL}_4(\\mathbb{Z})$ yield sharper bounds, which grow linearly or quadratically in explicit infinite families. For symmetric powers, combining our formulas with Horozov's calculation of the determinant-twisted summand yields exact identities and lower bounds for the untwisted summand, together with a conjectural degreewise description of its cohomology.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29796",
  "author_total": 2,
  "authors": [
    {
      "name": "Jitendra Bajpai"
    },
    {
      "name": "Taiwang Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0a5wj8nfqcza8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29796",
  "fulltext_url": null,
  "id": "0a5wj8nfqcza8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29796",
  "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.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7204958011",
  "pdf_url": "https://arxiv.org/pdf/2608.29796",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0a5wj8nfqcza8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29796",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29796"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Euler Characteristics of $\\mathrm{SL}_4(\\mathbb{Z})$ and $\\mathrm{GL}_4(\\mathbb{Z})$, and their cohomological consequences",
  "title_text": "Euler Characteristics of SL_4(ℤ) and GL_4(ℤ), and their cohomological consequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/0a5wj8nfqcza8",
  "venue": "arXiv",
  "versions": []
}