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On the diameters of McKay graphs for finite simple groups

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Abstract

Let G be a finite group, and α a nontrivial character of G. The McKay graph \({\cal M}\left({G,\alpha } \right)\) 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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Authors and Affiliations

  1. Department of Mathematics, Imperial College, London, SW7 2BZ, UK

    Martin W. Liebeck

  2. Institute of Mathematics, The Hebrew University of Jerrusalem Givat Ram, Jerusalem, 91904, Israel

    Aner Shalev

  3. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Pham Huu Tiep

Authors
  1. Martin W. Liebeck
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  2. Aner Shalev
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  3. Pham Huu Tiep
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Corresponding author

Correspondence to Pham Huu Tiep.

Additional 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.

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Liebeck, M.W., Shalev, A. & Tiep, P.H. On the diameters of McKay graphs for finite simple groups. Isr. J. Math. 241, 449–464 (2021). https://doi.org/10.1007/s11856-021-2109-1

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  • DOI: https://doi.org/10.1007/s11856-021-2109-1

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