# zbMATH — the first resource for mathematics

Turning weight multiplicities into Brauer characters. (English) Zbl 07203069
The purpose of the paper is to describe two methods for computing $$p$$-modular Brauer character tables for groups of Lie type $$G(p^f)$$ in defining characteristic $$p$$, assuming that the ordinary character table of $$G(p^f)$$ is known, and the weight multiplicities of the corresponding algebraic group G are known for $$p$$-restricted highest weights. As an application of these methods some character tables are computed for the modular ATLAS project.

##### MSC:
 20C20 Modular representations and characters 20C33 Representations of finite groups of Lie type
Full Text:
##### References:
 [1] Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, Computational Algebra and Number Theory. Computational Algebra and Number Theory, London, 1993. Computational Algebra and Number Theory. Computational Algebra and Number Theory, London, 1993, J. Symbolic Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039 [2] Brunat, O.; Lübeck, F., On defining characteristic representations of finite reductive groups, J. Algebra, 395, 121-141 (2013) · Zbl 1290.20011 [3] Breuer, T., CTblLib, the GAP character table library, Mar. 2013, GAP package [4] Carter, R. W., Finite Groups of Lie Type - Conjugacy Classes and Complex Characters (1985), A Wiley-Interscience Publication: A Wiley-Interscience Publication Chichester · Zbl 0567.20023 [5] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1985), Oxford University Press: Oxford University Press Eynsham, Maximal subgroups and ordinary characters for simple groups, with computational assistance from J.G. Thackray · Zbl 0568.20001 [6] GAP - Groups, Algorithms, and Programming (2018), Version 4.9.2 [7] Geck, M.; Hiss, G.; Lübeck, F.; Malle, G.; Pfeiffer, G., CHEVIE—a system for computing and processing generic character tables, Computational Methods in Lie Theory. Computational Methods in Lie Theory, Essen, 1994. Computational Methods in Lie Theory. Computational Methods in Lie Theory, Essen, 1994, Appl. Algebra Engrg. Comm. Comput., 7, 3, 175-210 (1996) · Zbl 0847.20006 [8] Jantzen, J. C., Representations of Algebraic Groups, Mathematical Surveys and Monographs (2003), American Mathematical Society [9] Jansen, C.; Lux, K.; Parker, R.; Wilson, R., An Atlas of Brauer Characters, London Mathematical Society Monographs. New Series, vol. 11 (1995), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, Appendix 2 by T. Breuer and S. Norton, Oxford Science Publications · Zbl 0831.20001 [10] Lübeck, F., Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math., 4, 135-169 (2001), (electronic) · Zbl 1053.20008 [11] Lübeck, F., Characters and Brauer trees of the covering group of $${}^2E_6(2)$$, (Finite Simple Groups: Thirty Years of the Atlas and Beyond. Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemp. Math., vol. 694 (2017), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 33-55 [12] Lübeck, F., Tables of weight multiplicities (2001-2018) [13] Parker, R. A., The computer calculation of modular characters (the meat-axe), (Atkinson, M. D., Computational Group Theory. Computational Group Theory, Durham, 1982 (1984), Academic Press: Academic Press London), 267-274 [14] Springer, T. A., Linear Algebraic Groups, Progress in Mathematics, vol. 9 (1998), Birkhäuser: Birkhäuser Boston · Zbl 0927.20024 [15] Steinberg, Robert, Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, vol. 80 (1968), American Mathematical Society: American Mathematical Society Providence, R.I. · Zbl 0164.02902 [16] Veldkamp, F. D., Representations of algebraic groups of type $$\operatorname{F}_4$$ in characteristic 2, J. Algebra, 16, 326-339 (1970) · Zbl 0215.11004 [17] Wilson, R. A., The modular atlas homepage (1998-2018) [18] Wilson, R. A., Atlas of Finite Group Representations [19] Wilson, R. A.; Parker, R. A.; Nickerson, S.; Bray, J. N.; Breuer, T., AtlasRep, a GAP interface to the Atlas of Group Representations (Jul. 2011), Refereed GAP package
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.