{
  "abstract": "We give two counterexamples to Stapledon's Effectiveness Conjecture. Each counterexample is a $6$-dimensional lattice polytope with an order-two symmetry whose equivariant $H^*$-series $φ[t]$ is a polynomial with one coefficient that is not effective. For polytopes with a symmetry of order two, the goal of this paper is to build up a picture of the landscape of results for when the Effectiveness Conjecture holds. Our main result is a criterion, in terms of the faces of $P$, that determines when the Ehrhart ring of $P$ admits an invariant linear system of parameters of the correct representation type. We use these results to reduce the space of possible counterexamples to one of two cases. A randomised computer search produced counterexamples in each case. We also give an example where the equivariant $H^*$-series is effective but admits no invariant linear system of parameters, neither for $P$ nor any invariant subdivision of it, answering a question of Stapledon in the case of groups of order two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17988",
  "author_total": 1,
  "authors": [
    {
      "name": "Oliver Clarke"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cbtkzfjf6xpfk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17988",
  "fulltext_url": null,
  "id": "cbtkzfjf6xpfk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17988",
  "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",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7221819855",
  "pdf_url": "https://arxiv.org/pdf/2610.17988",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cbtkzfjf6xpfk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17988",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17988"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II",
  "title_text": "The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/cbtkzfjf6xpfk",
  "venue": "arXiv",
  "versions": []
}