Problems on Brauer characters

Liu, Yanjun; Willems, Wolfgang; Zhang, Jiping

doi:10.22108/ijgt.2025.147244.1995

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	Problems on Brauer characters
International Journal of Group Theory
دوره 15، شماره 4، اسفند 2026، صفحه 179-192 اصل مقاله (456.71 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2025.147244.1995
نویسندگان
Yanjun Liu^* ¹؛ Wolfgang Willems²؛ Jiping Zhang³
¹School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, China
²Fakultat fur Mathematik, Otto-von-Guericke Universitat, Magdeburg, Germany and Departamento de Matematicas Universidad del Norte, Barranquilla, Colombia
³School of Mathematical Sciences, Peking University, Beijing, China
چکیده
In this paper we focus on problems of irreducible $p$-Brauer characters and state conjectures based on many examples which we computed. Many of the asked questions hold true for $p$-solvable groups, but their answers in general seem to require a much deeper understanding than we have at the moment. The questions are dealing with degrees, Hilbert divisors, divisibility, height-zero irreducible Brauer characters, number of irreducible Brauer characters in a $p$-block, and Cartan invariants. For instance, one of the main conjectures states that an irreducible Brauer character has Hilbert divisor 1 if and only if the character lies in a $p$-block of defect zero. This is true for $p$-solvable groups since in this case there is a relation between Hilbert divisors and vertices. However, even for a $p$-block of a non-$p$-solvable group containing only two irreducible Brauer characters we do not have an idea how to attack the problem. We hope that the conjectures and questions which we state in this paper will inspire further research.
کلیدواژه‌ها
Degree؛ $p$-block؛ height-zero Brauer character؛ Hilbert divisor؛ Cartan invariant

مراجع
[1] J. L. Alperin and L. Evans, Representations, resolutions, and Quillen’s dimension theorem, J. Pure Appl. Algebra, 22 no. 1 (1981) 1–9. [2] D. J. Benson, Representations and cohomology, II, Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics, 31, Cambridge University Press, Cambridge, 1991. [3] C. Bessenrodt and W. Willems, Relations between complexity and modular invariants and consequences for p-soluble groups, J. Algebra, 86 no. 2 (1984) 445–456. [4] R. Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol.I, Wiley, New York-London, (1963) 133–175. [5] W. Feit, The representation theory of finite groups, North-Holland Mathematical Library, 25, North-Holland Publishing Co., Amsterdam-New York, 1982. [6] W. Hammernik and G. Michler, On vertices of simple modules for p-solvable groups, Mitt. Math. Sem. Giessen Heft, 121 (1976) 147–162. [7] B. Huppert and N. Blackburn, Finite groups, II, Grundlehren der Mathematischen Wissenschaften, 242, Springer-Verlag, Berlin-New York, 1982. [8] P. G. Gres’, On conjectures of Olsson, Brauer and Alperin, Mat. Zametki, 52 no. 1 (1992) 32–35, 155; translation in Math. Notes, 52 no. 1-2 (1992) 654–657 (1993). [9] P. Landrock, A counterexample to a conjecture on the Cartan invariants of a group algebra, Bull. London Math. Soc., 5 (1973) 223–224. [10] Y. Liu and W. Willems, Quasi-projective Brauer characters, J. Algebra, 499 (2018) 506–515. [11] Y. Liu and W. Willems, On Hilbert divisors of Brauer characters, J. Algebra, 558 (2020) 595–610. [12] Y. Liu, W. Willems and H. Xiong, A generalization of Murai’s conjecture, Osaka J. Math., 61 no. 1 (2024) 107–119. [13] G. Malle, On Willems’ conjecture on Brauer character degrees, Adv. Math., 380 Paper No. 107609 (2021) 15 pp. [14] G. Malle, G. Navarro, A. A. Schaeffer Fry and P. Huu. Tiep, Brauer’s height zero conjecture, Annals of Math. (2), 200 no. 2 (2024) 557–608. [15] G. Malle and G. R. Robinson, On the number of simple modules in a block of a finite group, J. Algebra, 475 (2017) 423–438. [16] J. Maslowski, Evidenz zur Willemsschen Vermutung ¨uber Brauercharaktergrade, Diplomarbeit, Universit¨at Kassel, 2005. [17] G. O. Michler, Brauer’s conjectures and the classification of finte simple groups, Representation Theory II, Groups and Orders, Lecture Notes in Mathematics, Springer, Heidelberg, (1986) 129–142. [18] M. Murai, A note on the number of irreducible characters in a p-block of a finite group, Osaka J. Math., 21 no. 2 (1984) 387–398. [19] H. Nagao and Y. Tsushima, Representations of finite groups, Translated from the Japanese, Academic Press, Inc., Boston, MA, 1989. [20] G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, 250, Cambridge University Press, Cambridge, 1998. [21] G. Navarro, What do the modular characters know?, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 114 no. 1 Paper No. 15 (2020) 6 pp. [22] G. Navarro and B. Sambale, A counterexample to Feit’s problem VIII on decomposition numbers, J. Algebra, 477 (2017) 494–495. [23] G. Navarro, Problems on characters: solvable groups, Publ. Math., 67 no. 1 (2023) 173–198. [24] T. Okuyama, Module correspondence in finite groups, Hokkaido Math. J., 10 no. 3 (1981) 299–318. [25] J. B. Olsson, On 2-blocks with quaternion and quasidihedral defect groups, J. Algebra, 36 no. 2 (1975) 212–241. [26] B. Rothschild, Degrees of irreducible modular characters of blocks with cyclic defect groups, Bull. Amer. Math. Soc., 73 (1967) 102–104. [27] A. Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1986. [28] The GAP Group, GAP- Groups, algorithms, and programming, Version 4.15.1; 2025. (https://www.gap-system.org) [29] H. P. Tong-Viet, Some conjectures on Brauer character degrees, J. Algebra, 550 (2020) 210–218. [30] W. Willems, On degrees of irreducible Brauer characters, Trans. Amer. Math. Soc. 357 no. 6 (2005) 2379–2387. [31] Ch. Xu, K. Zhang and Y. Zhou, Hilbert divisors and degrees of irreducible Brauer characters, J. Group Theory, 28 no. 4 (2025) 971–985.
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Problems on Brauer characters