Abstract
Let \(G \) be a finite group of Lie type \(E_7 \) or \(E_8\) over a field \(\mathbb {F}_q \) and let \(W \) be the Weyl group of \(G \). In the present article, we find all maximal tori \(T \) of the group \(G \) that admit complements in the algebraic normalizer \(N(G,T) \). For every group under consideration except for the simply connected group \(E_7(q)\), we prove the following assertion: If \(w\in W\) and the corresponding torus \(T\) lacks the complement then there exists a lift of \(w\) in \(N(G,T) \) of order \(|w| \). In the exceptional case, we find all elements \(w \) admitting a lift in \(N(G,T) \) of order \(|w| \).
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REFERENCES
J. Adams and X. He, “Lifting of elements of Weyl groups,” J. Algebra 485, 142 (2017).
W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system. I: The user language,” J. Symbolic Comput. 24, 235 (1997).
A. A. Buturlakin and M. A. Grechkoseeva, “The cyclic structure of maximal tori of the finite classical groups,” Algebra i logika 46, 129 (2007) [Algebra and Logic 46, 73 (2007)].
R. W. Carter, “Conjugacy classes in the Weyl group,” Compositio Math. 25, 1 (1972).
R. W. Carter, Simple Groups of Lie Type (John Wiley and Sons, London, 1972).
R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (John Wiley and Sons, Chichester–New York, 1985).
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups (Clarendon Press, Oxford, 1985).
D. I. Deriziotis and A. P. Fakiolas, “The maximal tori in the finite Chevalley groups of type \( E_6\), \(E_7 \), and \(E_8 \),” Commun. Algebra 19, 889 (1991).
A. A. Galt, “On the splitting of the normalizer of a maximal torus in symplectic groups,” Izv. Ross. Akad. Nauk, Ser. Mat. 78, no. 3, 19 (2014) [Izv. Math. 78, 443 (2014)].
A. A. Galt, “On splitting of the normalizer of a maximal torus in linear groups,” J. Algebra Appl. 14, Art. 1550114 (2015).
A. A. Galt, “On splitting of the normalizer of a maximal torus in orthogonal groups,” J. Algebra Appl. 16, Art. 1750174 (2017).
A. A. Galt, “On the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups,” Izv. Ross. Akad. Nauk, Ser. Mat. 81, no. 2, 35 (2017) [Izv. Math. 81, 269 (2017)].
A. A. Galt and A. M. Staroletov, “On splitting of the normalizer of a maximal torus in \(E_6(q)\),” Algebra Colloq. 26, 329 (2019).
G. Lusztig, “Lifting involutions in a Weyl group to the torus normalizer,” Represent. Theory 22, 27 (2018).
The GAP Group, “GAP — Groups, Algorithms, and Programming, Version 4.9.1,” 2018 (https://www.gap-system.org).
J. Tits, “Normalisateurs de tores. I. Groupes de Coxeter étendus,” J. Algebra 4, 96 (1966).
N. A. Vavilov, “Do it yourself: The structure constants for Lie algebras of types \(E_l \),” Zap. Nauchn. Semin. POMI 281, 60 (2001) [J. Math. Sci. N.Y. 120, 1513 (2001)].
A. A. Galt and A. M. Staroletov, “Auxiliary calculations for maximal tori in \(E_7(q) \),” https://github.com/AlexeyStaroletov/GroupsOfLieType/blob/master/E7/complementsE7.txt.
A. A. Galt and A. M. Staroletov, “Auxiliary calculations for maximal tori in \(E_8(q) \),” https://github.com/AlexeyStaroletov/GroupsOfLieType/blob/master/E8/complementsE8.txt.
Computational Algebra Group, “Magma Calculator,” http://magma.maths.usyd.edu.au/calc.
ACKNOWLEDGMENTS
The authors are grateful to the anonymous referee for her/his valuable remarks.
Funding
The work was partially supported by the Russian Scientific Foundation (project no. 14-21-00065).
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Galt, A.A., Staroletov, A.M. On Splitting of the Normalizer of a Maximal Torus in \(E_7(q) \) and \(E_8(q) \). Sib. Adv. Math. 31, 244–282 (2021). https://doi.org/10.1134/S1055134421040027
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DOI: https://doi.org/10.1134/S1055134421040027