McKay graphs for alternating and classical groups

TL;DR

This paper investigates the diameters of McKay graphs for finite simple groups, proving bounds for alternating groups and classical groups, advancing understanding of their structural properties.

Contribution

It proves a conjecture on diameter bounds for McKay graphs of alternating groups and establishes linear bounds for classical groups of symplectic or orthogonal type.

Findings

Diameter bound for alternating groups proportional to log of group size over log of character degree.

Linear diameter bound for classical groups of symplectic or orthogonal type.

Confirmation of a conjecture regarding McKay graph diameters for alternating groups.

Abstract

Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The McKay graph $\mathcal{M}(G,\alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent of $\alpha\chi_1$. We study the diameters of McKay graphs for finite simple groups $G$. For alternating groups, we prove a conjecture made in [LST]: there is an absolute constant $C$ such that $\hbox{diam}\,{\mathcal M}(G,\alpha) \le C\frac{\log |\mathsf{A}_n|}{\log \alpha(1)}$ for all nontrivial irreducible characters $\alpha$ of $\mathsf{A}_n$. Also for classsical groups of symplectic or orthogonal type of rank $r$, we establish a linear upper bound $Cr$ on the diameters of all nontrivial McKay graphs.