{
  "abstract": "For each integer $l\\ge1$, we give closed formulas for the local energies on tensor squares of the level-$l$ perfect crystals of types $D_4^{(3)}$ and $G_2^{(1)}$, normalized to vanish at the all-zero coordinate tensors. The $D_4^{(3)}$ energy is the maximum of three defects, equivalently of nine linear forms in parameters of classical highest-weight elements. Eliminating these parameters yields the maximum of nine piecewise-linear functions of the twelve input coordinates. Tensor-compatible similarity identifies the $G_2^{(1)}$ energy at level $l$ with one third of the pullback of the $D_4^{(3)}$ energy at level $3l$, yielding the same coordinate formula. The formula has no explicit level dependence, respects natural coordinate-set inclusions between levels, and allows direct evaluation without the combinatorial $R$-matrix or crystal-graph traversal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02789",
  "author_total": 1,
  "authors": [
    {
      "name": "Shaolong Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b5t34vk9ab1vw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02789",
  "fulltext_url": null,
  "id": "b5t34vk9ab1vw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02789",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02789",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b5t34vk9ab1vw.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02789",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02789"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Closed energy formulas on tensor squares of perfect crystals of types $D_4^{(3)}$ and $G_2^{(1)}$",
  "title_text": "Closed energy formulas on tensor squares of perfect crystals of types D_4^(3) and G_2^(1)",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/b5t34vk9ab1vw",
  "venue": "arXiv",
  "versions": []
}