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arXiv · 2609.31026

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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Shengjun Zhang. 2026-09-25. Positivity Rigidity for Grossman-Larson Characters and the Kingman Face of the Hoffman Rooted-Tree Graph. https://arxiv.org/abs/2609.31026

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