McKay graphs for alternating and classical 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 $diam\mathcal{M}(G,\alpha )\le C\frac{\log |{\mathsf{A}}_n|}{\log \alpha (1)}$ 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.

22 pages