On the diameters of McKay graphs for finite simple groups

Martin W. Liebeck; Aner Shalev; Pham Huu Tiep

doi:10.1007/s11856-021-2109-1

On the diameters of McKay graphs for finite simple groups

Martin W. Liebeck, Aner Shalev, Pham Huu Tiep^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Let G be a finite group, and α a nontrivial character of G. The McKay graph 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 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 = PSL_n(q) or PSU_n(q).

Original language	American English
Pages (from-to)	449-464
Number of pages	16
Journal	Israel Journal of Mathematics
Volume	241
Issue number	1
DOIs	https://doi.org/10.1007/s11856-021-2109-1
State	Published - Mar 2021

Bibliographical 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) and the Joshua Barlaz Chair in Mathematics. The second and the third authors were also 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 in Berkeley, California, during the Spring 2018 semester.

Publisher Copyright:
© 2021, The Hebrew University of Jerusalem.

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abstract = "Let G be a finite group, and α a nontrivial character of G. The McKay graph 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 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 = PSLn(q) or PSUn(q).",

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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) and the Joshua Barlaz Chair in Mathematics. The second and the third authors were also 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 in Berkeley, California, during the Spring 2018 semester. Publisher Copyright: {\textcopyright} 2021, The Hebrew University of Jerusalem.",

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N2 - Let G be a finite group, and α a nontrivial character of G. The McKay graph 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 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 = PSLn(q) or PSUn(q).

AB - Let G be a finite group, and α a nontrivial character of G. The McKay graph 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 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 = PSLn(q) or PSUn(q).

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On the diameters of McKay graphs for finite simple groups

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