@article{497e2872eb9c4e5088c94ea7e82d11f9,

title = "Mckay graphs for alternating and classical groups",

abstract = "Let G be a finite group, and α a nontrivial character of G. The McKay graphM(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 finite simple groups G. For alternating groups G = An, we prove a conjecture made in another work by the authors: There is an absolute constant C such that diamM(G, α) ≤ C log |G| log α(1) for all nontrivial irreducible characters α 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 χ1χ2 χl of irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents.",

keywords = "McKay graph, Simple groups",

author = "LIEBECK, {MARTIN W.} and ANER SHALEV and TIEP, {PHAM HUU}",

note = "Funding Information: The second author acknowledges the support of ISF grant 686/17, and the Vinik chair of mathematics which he holds. The third author gratefully acknowledges the support of the NSF (grant DMS-1840702), the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at the Institute for Advanced Study (Princeton, NJ). The second and the third authors were partially supported by BSF grant 2016072. The authors also acknowledge the support of the National Science Foundation under Grant No. DMS-1440140 while they were in residence at the Mathematical Sciences Research Institute (Berkeley, CA), during the Spring 2018 semester. Part of this work was done when the authors were in residence at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) in Spring 2020, and partially supported by a grant from the Simons Foundation. Publisher Copyright: {\textcopyright} 2021 American Mathematical Society.",

year = "2021",

doi = "10.1090/tran/8395",

language = "English (US)",

volume = "374",

pages = "5651--5676",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "8",

}