On the diameters of McKay graphs for finite simple groups

2019/11/29 by Liebeck, Martin W., Shalev, Aner, Tiep, Pham Huu
#20C15 #20C30 #20C33 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1911.13284

Abstract

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 χ₁ to χ₂ if χ₂ is a constituent of αχ₁. We study the diameters of McKay graphs for simple groups G. For G a group of Lie type, we show that for any α, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for G=\rm PSL_n(q) or \rm PSU_n(q). We also bound the diameter for symmetric and alternating groups.

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