Semifinite harmonic functions on the Gnedin-Kingman graph

TL;DR

This paper classifies indecomposable semifinite harmonic functions on the Gnedin-Kingman graph, linking combinatorial structures with algebraic properties of quasisymmetric functions, and establishes a multiplicativity property analogous to the Vershik-Kerov ring theorem.

Contribution

It provides a detailed classification of harmonic functions on the Gnedin-Kingman graph and proves a multiplicativity property similar to the Vershik-Kerov ring theorem.

Findings

Classification of indecomposable semifinite harmonic functions

Establishment of a multiplicativity property

Connection to the Vershik-Kerov ring theorem

Abstract

We study the Gnedin-Kingman graph, which corresponds to Pieri's rule for the monomial basis $\{M_{\lambda}\}$ in the algebra $\mathrm{QSym}$ of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin-Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik-Kerov ring theorem.