{
  "abstract": "We study torus-orbit closures in the type $A$ complete flag variety naturally associated to cones in the positive cluster fan, together with their left $S_n$-translates. The torus-equivariant degree maps can be computed via composites of long-range divided difference operations encoded by noncrossing alternating forests, and we give combinatorial algorithms to expand the torus-equivariant homology classes into Graham-positive combinations of Schubert cycles. As applications we obtain combinatorial Graham-positive Schubert cycle expansions for all torus-invariant curves (generalizing the AJS-Billey formula for torus-fixed points), generic torus-orbit closures, and Richardson varieties for Bruhat intervals $[w,wc']$ where $c'\\le s_{n-1}s_{n-2}\\cdots s_1$. Projecting to Grassmannians we also obtain Graham-positive Grassmannian Schubert cycle decompositions of torus-orbit closures associated to lattice path matroids on permuted ground sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10735",
  "author_total": 2,
  "authors": [
    {
      "name": "Hunter Spink"
    },
    {
      "name": "Vasu Tewari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ryatvnt6gns2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10735",
  "fulltext_url": null,
  "id": "5ryatvnt6gns2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10735",
  "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"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212315448",
  "pdf_url": "https://arxiv.org/pdf/2609.10735",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ryatvnt6gns2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10735",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10735"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Long range divided differences, clusters, and Graham-positivity",
  "title_text": "Long range divided differences, clusters, and Graham-positivity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/5ryatvnt6gns2",
  "venue": "arXiv",
  "versions": []
}