×

The Brauer trees of the exceptional Chevalley groups of type \(E_ 6\). (English) Zbl 0846.20017

The Brauer trees for the exceptional Chevalley groups of type \(E_6\) are determined. It is an extension of the work of M. Geck [Commun. Algebra 20, No. 10, 2937-2973 (1992; Zbl 0770.20020)]. In order to solve this problem the authors have to consider the tensor products of uniform functions.

MSC:

20C33 Representations of finite groups of Lie type
20C20 Modular representations and characters
20G40 Linear algebraic groups over finite fields
20G05 Representation theory for linear algebraic groups

Citations:

Zbl 0770.20020

Software:

CHEVIE; CAS; GAP; Maple
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] M. Broué, Isométries parfaites, types de blocs, catégories dérivées,Astérisque 181–182 (1990), 61–92
[2] M. Broué, Isométries de caractères et equivalences de Morita ou dérivées,Publ. Math. I.H.E.S. 71 (1990), 45–63 · Zbl 0727.20005
[3] M. Broué, G. Malle andJ. Michel, Generic blocks of finite reductive groups,Astérisque 212 (1993), 7–92 · Zbl 0843.20012
[4] M. Broué andJ. Michel, Blocs à groupes de défaut abéliens des groupes réductifs finis,Astérisque 212 (1993), 93–117
[5] M. Cabanes and M. Enguehard, Local methods for the blocks of the finite reductive groups, Preprint · Zbl 0923.20035
[6] R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, New York, 1985 · Zbl 0567.20023
[7] B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan, and S. M. Watt, Maple V, Language Reference Manual, Springer-Verlag, 1991
[8] D. I. Deriziotis, Conjugacy classes and centralizers of semisimple elements in finite groups of Lie type, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, Heft 11, Germany, 1984 · Zbl 0574.20035
[9] F. Digne and J. Michel, Representations of Finite Groups of Lie Type, London Math. Soc. Student Texts 21, Cambridge University Press, 1991 · Zbl 0815.20014
[10] R. Dipper, On quotients of Hom-functors and representations of finite general linear groups I,J. Algebra 130 (1990), 235–259 · Zbl 0698.20005 · doi:10.1016/0021-8693(90)90111-Z
[11] W. Feit, Possible Brauer trees,Illinois J. Math. 28 (1984), 43–56 · Zbl 0538.20006
[12] B. Fischer, The character table ofE 6(2), private communication
[13] P. Fong andB. Srinivasan, Brauer trees inGL(n, q), Math. Z. 187 (1984), 81–88 · Zbl 0545.20006 · doi:10.1007/BF01163168
[14] P. Fong andB. Srinivasan, Brauer trees in classical groups,J. Algebra 131 (1990), 179–225 · Zbl 0704.20011 · doi:10.1016/0021-8693(90)90172-K
[15] M. Geck, Generalized Gelfand-Graev characters for Steinberg’s triality groups and their applications,Comm. Algebra 19 (1991), 3249–3269 · Zbl 0756.20001 · doi:10.1080/00927879108824318
[16] M. Geck, Brauer trees of Hecke algebras,Comm. Algebra 20 (1992), 2937–2973 · Zbl 0770.20020 · doi:10.1080/00927879208824499
[17] M. Geck, G. Hiss, F. Lübeck, G. Malle and G. Pfeiffer, CHEVIE–Generic Character Tables of Finite Groups of Lie Type, Hecke Algebras and Weyl Groups, Preprint 93-62, IWR der Universität Heidelberg, 1993
[18] G. Hiss, Regular and semisimple blocks of finite reductive groups,J. London Math. Soc. 41 (1990), 63–68 · Zbl 0661.20030 · doi:10.1112/jlms/s2-41.1.63
[19] G. Hiss, The Brauer trees of the Ree groups,Comm. Algebra 19 (1991), 871–888 · Zbl 0798.20010 · doi:10.1080/00927879108824175
[20] G. Hiss andK. Lux, Brauer Trees of Sporadic Groups, Oxford University Press, Oxford, 1989 · Zbl 0685.20013
[21] G. D. James, The representation theory of the symmetric groups, Springer Lecture Notes in Mathematics Vol. 682, Springer-Verlag, 1978 · Zbl 0393.20009
[22] C. Jansen, K. Lux, R. A. Parker, and R. A. Wilson, An Atlas of Modular Character Tables, to appear · Zbl 0831.20001
[23] F. Lübeck, Charaktertafeln für die Gruppen CSp6(q) mit ungerademq und SP6(q) mit gerademq, Dissertation, Heidelberg, 1993
[24] G. Lusztig, Coxeter orbits and eigenspaces of Frobenius,Invent. Math. 38 (1976), 101–159 · Zbl 0366.20031 · doi:10.1007/BF01408569
[25] J. Neubüser, H. Pahlings and W. Plesken, CAS; Design and use of a system for the handling of characters of finite groups,in: Computational Group Theory, Academic Press, 1984, pp. 195–247
[26] M. Schönert et al., GAP–Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, RWTH Aachen, Germany, fourth ed., 1994
[27] J. Shamash, Brauer trees for blocks of cyclic defect in the groupsG 2(q) for primes dividingq 2{\(\pm\)}q+1,J. Algebra 123 (1989), 378–396 · Zbl 0674.20004 · doi:10.1016/0021-8693(89)90052-5
[28] J. Shamash, Blocks and Brauer trees in the groupsG 2(q) for primes dividingq{\(\pm\)}1,Comm. Algebra 17 (1989), 1901–1949 · Zbl 0686.20006 · doi:10.1080/00927878908823827
[29] J. Shamash, Blocks and Brauer trees for the groupsG 2(2 k ),G 2(3 k ),Comm. Algebra 20 (1992), 1375–1387 · Zbl 0770.20011 · doi:10.1080/00927879208824409
[30] D. L. White, Brauer trees of 2.F 4(2),Comm. Algebra 20 (1992), 3353–3368 · Zbl 0808.20018 · doi:10.1080/00927879208824519
[31] E. Wings, Über die unipotenten Charaktere der Chevalley-Gruppen vom TypeF 4 in guter Charakteristik, Dissertation, Lehrstuhl D für Mathematik, RWTH Aachen, 1995 · Zbl 0858.20010
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.