Electrical Networks and Symplectic Invariants

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 Sp(2n)-invariant tensor Z_G. For Sp(2)=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 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 Sp(4)-invariant tensors. The coefficients of Z_G in this basis are weighted counts of reduced double groves, up to normalization.

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BibTeX RIS

BibTeX

@misc{kogan2026electrical,
  title = {{Electrical Networks and Symplectic Invariants}},
  author = {David Kogan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07539},
  eprint = {2610.07539},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07539},
}

RIS

TY  - UNPB
TI  - Electrical Networks and Symplectic Invariants
AU  - David Kogan
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07539
UR  - https://arxiv.org/abs/2610.07539
AB  - 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 Sp(2n)-invariant tensor Z_G. For Sp(2)=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 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 Sp(4)-invariant tensors. The coefficients of Z_G in this basis are weighted counts of reduced double groves, up to normalization.
ER  -

Abstract

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