On the diameters of McKay graphs for finite simple groups

TL;DR

This paper investigates the diameters of McKay graphs for finite simple groups, providing bounds that depend on the group's type and rank, with specific results for classical groups and symmetric groups.

Contribution

It establishes new bounds on the diameters of McKay graphs for various families of finite simple groups, including Lie type and symmetric groups.

Findings

Diameter bounded by quadratic function of rank for Lie type groups

Stronger bounds obtained for PSL_n(q) and PSU_n(q)

Diameter bounds for symmetric and 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 simple groups $G$. For $G$ a group of Lie type, we show that for any $\alpha$, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for $G={\rm PSL}_n(q)$ or ${\rm PSU}_n(q)$. We also bound the diameter for symmetric and alternating groups.