On the diameters of McKay graphs for finite simple 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 $G={\mathrm{PSL}}_n(q)$ or ${\mathrm{PSU}}_n(q)$. We also bound the diameter for symmetric and alternating groups.