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PreprintarXiv licencearXiv

Closed energy formulas on tensor squares of perfect crystals of types D4(3) and G2(1)

  • Shaolong Han
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Abstract

For each integer l≥1, we give closed formulas for the local energies on tensor squares of the level-l perfect crystals of types D4(3) and G2(1), normalized to vanish at the all-zero coordinate tensors. The D4(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 G2(1) energy at level l with one third of the pullback of the D4(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.

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Cite

BibTeXRIS
BibTeX
@misc{han2026closed,
  title = {{Closed energy formulas on tensor squares of perfect crystals of types $D_4^(3)$ and $G_2^(1)$}},
  author = {Shaolong Han},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02789},
  eprint = {2611.02789},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02789},
}
RIS
TY  - UNPB
TI  - Closed energy formulas on tensor squares of perfect crystals of types D_4^(3) and G_2^(1)
AU  - Shaolong Han
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02789
UR  - https://arxiv.org/abs/2611.02789
AB  - For each integer l≥1, 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.
ER  -