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).
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Z. Arad, D. Chillag and M. Herzog, Powers of characters of finite groups, Journal of Algebra 103 (1986), 241–255.
R. Bezrukavnikov, M. W. Liebeck, A. Shalev and P. H. Tiep, Character bounds for finite groups of Lie type, Acta Mathematica 221 (2018), 1–57.
R. Brauer, A note on theorems of Burnside and Blichfeldt, Proceedings of the American Mathematical Society 15 (1964), 31–34.
R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Pure and Applied Mathematics (New York), Wiley Interscience, New York, 1985.
J. Fulman and R. M. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Transactions of the American Mathematical Society 364 (2012), 3023–3070.
D. Gluck, Sharper character value estimates for groups of Lie type, Journal of Algebra 174 (1995), 229–266.
G. Heide, J. Saxl, P. H. Tiep and A. E. Zalesski, Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type, Proceedings of the London Mathematical Society 106 (2013), 908–930.
M. Larsen, G. Malle and P. H. Tiep, The largest irreducible representations of simple groups, Proceedings of the London Mathematical Society 106 (2013), 65–96.
M. W. Liebeck and A. Shalev, Diameters of simple groups: sharp bounds and applications, Annals of Mathematics 154 (2001), 383–406.
M. W. Liebeck and A. Shalev, Fuchsian groups, finite simple groups, and representation varieties, Inventiones Mathematicae 159 (2005), 317–367.
J. McKay, Graphs, singularities and finite groups, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proceedings of Symposia in Pure Mathematics, Vol. 37, American mathematical Society, Providence, RI, 1980, pp. 183–186.
J. Taylor and P. H. Tiep, Lusztig induction, unipotent supports, and character bounds, Transactions of the American Mathematical Society 373 (2020), 8637–8676.
P. H. Tiep and A. E. Zalesskii, Minimal characters of the finite classical groups, Communications in Algebra 24 (1996), 2093–2167.
P. H. Tiep and A. E. Zalesskii, Some characterizations of the Weil representations of the symplectic and unitary groups, Journal of Algebra 192 (1997), 130–165.
I. Zisser, The character covering number for alternating groups, Journal of Algebra 153 (1992), 357–372.
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-021-2109-1