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Positivity Rigidity for Grossman-Larson Characters and the Kingman Face of the Hoffman Rooted-Tree Graph

  • Shengjun Zhang
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Abstract

We classify the real characters of the Grossman-Larson Hopf algebra that are nonnegative on the rooted-tree basis. They vanish on trees with branching away from the root, and the normalized characters are parametrized by Kingman paintboxes. Two-sided Pieri states are mixtures of the normalized characters, with unique mixing measures. The corresponding laws form a proper exposed Bauer face of the simplex of central measures on the Hoffman rooted-tree graph. For the path-forest subgraph, the full and minimal Martin boundaries coincide and are homeomorphic to the Kingman simplex. The limiting ranked root-branch frequencies generate the completed central tail.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026positivity,
  title = {{Positivity Rigidity for Grossman-Larson Characters and the Kingman Face of the Hoffman Rooted-Tree Graph}},
  author = {Shengjun Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31026},
  eprint = {2609.31026},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31026},
}
RIS
TY  - UNPB
TI  - Positivity Rigidity for Grossman-Larson Characters and the Kingman Face of the Hoffman Rooted-Tree Graph
AU  - Shengjun Zhang
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31026
UR  - https://arxiv.org/abs/2609.31026
AB  - We classify the real characters of the Grossman-Larson Hopf algebra that are nonnegative on the rooted-tree basis. They vanish on trees with branching away from the root, and the normalized characters are parametrized by Kingman paintboxes. Two-sided Pieri states are mixtures of the normalized characters, with unique mixing measures. The corresponding laws form a proper exposed Bauer face of the simplex of central measures on the Hoffman rooted-tree graph. For the path-forest subgraph, the full and minimal Martin boundaries coincide and are homeomorphic to the Kingman simplex. The limiting ranked root-branch frequencies generate the completed central tail.
ER  -