Traces On Diagram Algebras I: Free Partition Quantum Groups, Random Lattice Paths And Random Walks On Trees

TL;DR

This paper classifies extremal traces on certain algebras linked to free partition quantum groups, connecting them to random lattice paths and walks on trees, and provides new formulas for their irreducible representation dimensions.

Contribution

It offers the first classification of extremal traces for these algebras, computes minimal boundaries for random walks on Fibonacci trees, and derives new formulas for representation dimensions.

Findings

Classification of extremal traces on seven algebras.

Solution of the minimal boundary for Fibonacci tree walks.

New formulas for irreducible representation dimensions.

Abstract

We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica-Speicher (arXiv:0808.2628) and Weber (arXiv:1201.4723). For the infinite-dimensional Temperley-Lieb-algebra (corresponding to the quantum group $O^+_N$) and the Motzkin algebra ($B^+_N$), the classification of extremal traces implies a classification result for well-known types of central random lattice paths. For the $2$-Fuss-Catalan algebra ($H_N^+$) we solve the classification problem by computing the \emph{minimal or exit boundary} (also known as the \emph{absolute}) for central random walks on the Fibonacci tree, thereby solving a probabilistic problem of independent interest, and to our knowledge the first such result for a nonhomogeneous tree. In the course of this article, we also discuss the branching graphs for all seven examples of free partition quantum groups, compute those that were not already known, and provide new formulas for the dimensions of their irreducible representations.