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

Sugimoto, Masahiro

doi:10.22108/ijgt.2024.140888.1896

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	Harada's conjecture II for the finite general linear groups and unitary groups
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 21 آبان 1403 اصل مقاله (473.22 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2024.140888.1896
نویسنده
Masahiro Sugimoto^*
Department of Mathematics, University of Tsukuba 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577 Japan
چکیده
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.
کلیدواژه‌ها
irreducible character؛ conjugacy classes؛ partitions

مراجع
[1] T. Abe and N. Chigira, Towards a solution to Harada’s conjecture II, RIMS Kokyuroku, 2189 (2021) 77–86. [2] V. Ennola,On the characters of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I, 323 (1963) 35 pp. [3] V. Ennola, On the conjugacy classes of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I, 313 (1962) 13 pp. [4] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.12.2, 2022, https://www.gap-system.org [5] J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc., 80 (1955) 402–447. [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.
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Harada's conjecture II for the finite general linear groups and unitary groups