交替群和经典群的McKay图

M. Liebeck, A. Shalev, P. Tiep
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引用次数: 3

摘要

设$G$是一个有限群,$\alpha$是$G$的一个非平凡特征。McKay图$\mathcal{M}(G,\alpha)$以$G$的不可约特征为顶点,如果$\chi_2$是$\alpha\chi_1$的一个组成部分,则有一条从$\chi_1$到$\chi_2$的边。我们研究了有限简单群的McKay图的直径$G$。对于交替群,我们证明了[LST]中的一个猜想:存在一个绝对常数$C$,使得$\hbox{diam}\,{\mathcal M}(G,\alpha) \le C\frac{\log |\mathsf{A}_n|}{\log \alpha(1)}$对于$\mathsf{A}_n$的所有非平凡不可约字符$\alpha$。对于秩为$r$的辛型或正交型的经典群,我们在所有非平凡McKay图的直径上建立了一个线性上界$Cr$。
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McKay graphs for alternating and classical groups
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.
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