McKay graphs for alternating and classical groups

Liebeck, Martin; Shalev, Aner; Tiep, Pham

doi:10.1090/tran/8395

McKay graphs for alternating and classical groups
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by Martin W. Liebeck, Aner Shalev and Pham Huu Tiep PDF

Trans. Amer. Math. Soc. 374 (2021), 5651-5676 Request permission

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 $G = \mathsf {A}_n$, we prove a conjecture made in another work by the authors: there is an absolute constant $C$ such that $\mathrm {diam} {\mathcal M}(G,\alpha ) \le C\frac {\log |G|}{\log \alpha (1)}$ for all nontrivial irreducible characters $\alpha$ of $G$. Also for classical groups of symplectic or orthogonal type of rank $r$, we establish a linear upper bound $Cr$ on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a product $\chi _1\chi _2\cdots \chi _l$ of irreducible characters of some finite simple groups $G$ to contain all irreducible characters of $G$ as constituents.

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