{
  "abstract": "Let $P$ be a finite connected poset and let $Λ_P$ be the opposite endomorphism algebra of the direct sum of all interval representations of $P$ over a field. Via projectivization, this algebra governs resolutions relative to interval-decomposable representations, which arise naturally in persistence theory. We first show that $Λ_P$ carries a Reedy algebra structure in the sense of Dalezios--Šťov\\'ıček. Its Reedy degree is given by the cardinality of the indexing interval, and the induced quasi-hereditary order is given by reverse interval cardinality. With respect to the resulting quasi-hereditary structure, we give a concrete combinatorial description of the standard modules and construct explicit projective resolutions of these modules. Using these resolutions, we reduce the calculation of standard--simple Ext groups to the reduced cohomology of simplicial complexes determined by the interval combinatorics. Order reversal gives the corresponding simple--costandard formula. Building on these calculations, we determine all simple--simple Ext groups. These groups are one-dimensional in a unique degree when the corresponding pair of intervals is saturated, and vanish otherwise. As a consequence, we obtain an exact combinatorial formula for the global dimension of $Λ_P$, which in particular shows that it is independent of the coefficient field. As an application, for the $m$ by $\\ell$ grid $G_{m,\\ell}$ with $m\\geq\\ell\\geq2$, we give the explicit formula $\\operatorname{gldim}Λ_{G_{m,\\ell}}=\\min\\{2\\ell,m+\\ell-2\\}$. This also gives an explicit formula for the interval-resolution global dimension of these grids, settling the corresponding grid conjectures of Asashiba--Escolar--Nakashima--Yoshiwaki and determining the stable value and the precise stabilization threshold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15927",
  "author_total": 1,
  "authors": [
    {
      "name": "Toshitaka Aoki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ezzg76hh530r5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15927",
  "fulltext_url": null,
  "id": "ezzg76hh530r5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15927",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7213335874",
  "pdf_url": "https://arxiv.org/pdf/2609.15927",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ezzg76hh530r5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15927",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15927"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Interval endomorphism algebras of posets: Reedy structure, combinatorics, and homological theory",
  "title_text": "Interval endomorphism algebras of posets: Reedy structure, combinatorics, and homological theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/ezzg76hh530r5",
  "venue": "arXiv",
  "versions": []
}