Semifinite harmonic functions on the zigzag graph

TL;DR

This paper classifies indecomposable semifinite harmonic functions on the zigzag graph, linking them to Pieri's rule for quasisymmetric functions and establishing an analog of the Vershik-Kerov ring theorem.

Contribution

It provides a complete classification and explicit construction of semifinite harmonic functions on the zigzag graph, extending the Vershik-Kerov ring theorem.

Findings

Classification parameters for indecomposable semifinite harmonic functions

Explicit construction method for these functions

Semifinite analog of the Vershik-Kerov ring theorem

Abstract

We study semifinite harmonic functions on the zigzag graph, which corresponds to Pieri's rule for the fundamental quasisymmetric functions $\{F_{\lambda}\}$. The main problem, which we solve here, is to classify the indecomposable semifinite harmonic functions on this graph. We describe the set of classification parameters and an explicit construction that produces a semifinite indecomposable harmonic function out of every point of this set. We also establish a semifinite analog of the Vershik-Kerov ring theorem.