{
  "abstract": "Consider a finite planar graph with positive real edge weights and designated boundary vertices, called nodes. Such a graph is called a circular planar electrical network. A grove is a spanning forest in which every component contains at least one node. The connected components of a grove determine a partition of the nodes. We relate weighted grove counts to invariant theory for the symplectic group. To a planar electrical network $G$, we associate an $\\mathrm{Sp}(2n)$-invariant tensor $Z_G$. For $\\mathrm{Sp}(2)=\\mathrm{SL}(2)$, we expand $Z_G$ in the Temperley--Lieb basis indexed by noncrossing matchings and relate its coefficients to the Kenyon--Wilson grove formulas. For $\\mathrm{Sp}(4)$, we give reduction rules for superpositions of two groves and prove that the tensors indexed by $3$-noncrossing matchings form a basis of the space of $\\mathrm{Sp}(4)$-invariant tensors. The coefficients of $Z_G$ in this basis are weighted counts of reduced double groves, up to normalization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07539",
  "author_total": 1,
  "authors": [
    {
      "name": "David Kogan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/49dapgn6kdvaj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07539",
  "fulltext_url": null,
  "id": "49dapgn6kdvaj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07539",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221019905",
  "pdf_url": "https://arxiv.org/pdf/2610.07539",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/49dapgn6kdvaj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2610.07539",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07539"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Electrical Networks and Symplectic Invariants",
  "title_text": "Electrical Networks and Symplectic Invariants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/49dapgn6kdvaj",
  "venue": "arXiv",
  "versions": []
}