交错群和典型群的McKay图
McKay graphs for alternating and classical groups
摘要：
设$G$是有限群，$α$是$G$的非平凡特征标. McKay图$\mathcal{M}(G,α)$以$G$的不可约特征标为顶点，如果$χ_2$是$αχ_1$的组成部分，则有一条从$χ_1$到$χ_2$的边。研究了有限单群$G$的McKay图的直径.对于交错群，我们证明了[LST]中的一个猜想：存在一个绝对常数$C$，使得对于$\mathsf{A}_n$的所有非平凡不可约特征标$α$，有$\hbox{diam}\,{\mathcal M}(G,α) \le C\frac{\log |\mathsf{A}_n|}{\log α(1)}$.对于秩为$r$的辛型或正交型的经典群，我们建立了所有非平凡McKay图的直径的线性上界$Cr$.
英文摘要：
Let $G$ be a finite group, and $α$ a nontrivial character of $G$. The McKay graph $\mathcal{M}(G,α)$ has the irreducible characters of $G$ as vertices, with an edge from $χ_1$ to $χ_2$ if $χ_2$ is a constituent of $αχ_1$. 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 $\hbox{diam}\,{\mathcal M}(G,α) \le C\frac{\log |\mathsf{A}_n|}{\log α(1)}$ for all nontrivial irreducible characters $α$ 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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