Special characters on star graphs and representations of $*$-algebras

TL;DR

This paper investigates special characters on star graphs, analyzing their properties and evolution, and demonstrates that associated $*$-algebras possess irreducible infinite-dimensional representations when extended Dynkin graphs are involved.

Contribution

It introduces a new class of special characters on star graphs and establishes their role in the representation theory of $*$-algebras, especially in relation to extended Dynkin subgraphs.

Findings

Special characters can be decomposed into odd and even parts.

Formulas for character evolution under reflections are derived.

$*$-algebras have irreducible infinite-dimensional representations when extended Dynkin graphs are present.

Abstract

For a star-shaped graph, we introduce special characters and study their properties. We decompose special characters into odd and even parts and study their evolution under reflections. We apply the obtained formulas to prove that the corresponding $*$-algebra have irreducible infinite-dimensional $*$-representations, if the graph contains an extended Dynkin graph as a proper subgraph.