Harada's conjecture II for the finite general linear groups and unitary groups

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, University of Tsukuba 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577 Japan

Abstract

K. Harada conjectured for any finite group $G$, the product of sizes of all conjugacy classes is divisible by the product of degrees of all irreducible characters. We study this conjecture when $G$ is the general linear group over a finite field. We show the conjecture holds if the order of the field is sufficiently large.

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References

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[3] V. Ennola, On the conjugacy classes of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I, 313 (1962) 13 pp.
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[6] K. Harada, Revisiting character theory of finite groups, Bull. Inst. Math. Acad. Sin. (N.S.), 13 no. 4 (2018) 383–395.
[7] A. Hida, Harada’s conjecture on character degrees and class sizes –symmetric and alternating groups–, RIMS Kokyuroku, 2086 (2018) 144–153.
[8] N. Kawanaka, Generalized Gelfand-Graev representations and Ennola duality, Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., North-Holland, Amsterdam, 1985 175–206.
International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 11 November 2024

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History

  • Receive Date: 17 March 2024
  • Revise Date: 20 October 2024
  • Accept Date: 06 November 2024
  • Published Online: 11 November 2024

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