Subgroups generated by a pair of 2-tori in GL(4,𝐾), III
https://doi.org/10.22405/2226-8383-2025-26-4-149-173
Abstract
In the present paper we complete the description of the subgroups generated by a pair of 2-tori in GL(𝑛,𝐾). Recall that 2-tori in GL(𝑛,𝐾) are the subgroups conjugate to the diagonal
subgroup of the following form diag(𝜀, 𝜀, 1, . . . , 1). In work [2] the reduction theorem for the pairs of 𝑚-tori was proved. It follows from it that any pair of 2-tori can be embedded in
GL(6,𝐾) by simultaneous conjugation. The orbit of a pair of 2-tori (𝑋, 𝑌 ) is called the orbit in GL(𝑛,𝐾), if the pair (𝑋, 𝑌 ) is embedded in GL(𝑛,𝐾) by simultaneous conjugation and it can not be embedded in GL(𝑛 − 1,𝐾). It is clear that 𝑛 can take values 3, 4, 5 and 6. In the same work the orbits and spans of 2-tori in GL(6,𝐾) were described. In the subsequent papers we described the pairs of 2-tori in GL(5,𝐾), the orbits of pairs of 2-tori in GL(4,𝐾) and the spans in GL(4,𝐾) corresponding to degenerate cases (the reductive part of the group is not larger than GL(2,𝐾)). In this paper we describe undegenerate cases of pairs of 2-tori in GL(4,𝐾). Thus we complete our description. The most difficult subgroups turns out the groups with a reductive part SL(2,𝐾) × SL(2,𝐾) or SL(2,𝐿), where [𝐿 : 𝐾] = 2.
About the Authors
Vladimir Vicktorovich Nesterov
Saint Petersburg State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Meiling Zhang
Saint Petersburg State University
Russian Federation
Russian Federation
postgraduate student
References
1. Cohen, A.M., Cuypers, H., and Sterk, H., 1999, “Linear groups generated by reflection tori”, Canadian Journal of Mathematics, 51(6), pp. 1149–1174.
2. Nesterov, V.V., and Vavilov, N.A., 2020, “Pairs of microweight tori in GL𝑛”, Chebyshevskii Sbornik, 21(3), pp. 256–266.
3. Nesterov, V.V., and Zhang, M., 2024, “Subgroups generated by a pair of 2-tori in GL(4,𝐾). I”, Zapiski Nauchnykh Seminarov POMI, 531, pp. 127–146.
4. Nesterov, V.V., and Zhang, M., 2025, “Subgroups generated by a pair of 2-tori in GL(4,𝐾). II”, Vladikavkaz Mathematical Journal, 27(3), pp. 101–119.
5. Vavilov, N.A., 1990, “Subgroups of Chevalley groups that contain a maximal torus”, Trudy Leningradskogo Matematicheskogo Obshchestva, 1, pp. 64–109.
6. Vavilov, N.A., 2008, “Geometry of 1-tori in GL𝑛”, St. Petersburg Mathematical Journal, 19(3), pp. 407–429.
7. Vavilov, N.A., 2009, “Weight elements of Chevalley groups”, St. Petersburg Mathematical Journal, 20(1), pp. 23–57.
8. Vavilov, N.A., and Nesterov, V.V., 2008, “Geometry of microweight tori”, Vladikavkaz Mathematical Journal, 10(1), pp. 10–23.
9. Vavilov, N.A., and Nesterov, V.V., 2023, “Subgroups generated by a pair of 2-tori in GL(5,𝐾)”, Zapiski Nauchnykh Seminarov POMI, 522, pp. 8–45.
10. Vavilov, N.A., and Semenov, A.A., 1991, “Bruhat decomposition for long root tori in Chevalley groups”, Journal of Soviet Mathematics, 57(6), pp. 3453–3458.
11. Vavilov, N.A., and Semenov, A.A., 2013, “Long root tori in Chevalley groups”, St. Petersburg Mathematical Journal, 24(3), pp. 22–83.
12. Nesterov, V.V., 2005, “Generation of pairs of short root subgroups in Chevalley groups”, St. Petersburg Mathematical Journal, 16(6), pp. 1051–1077.
13. Nesterov, V.V., 2022, “Extraction of small rank unipotent elements in GL(4,𝐾)”, Journal of Mathematical Sciences, 264(1), pp. 86–95.
14. Shafarevich, I.R., and Remizov, A.O., 2013, Linear Algebra and Geometry, Berlin: Springer.
15. Hahn, A.J., and O’Meara, O.T., 1989, The Classical Groups and K-Theory, Berlin: Springer.
Review
For citations:
Nesterov V.V., Zhang M. Subgroups generated by a pair of 2-tori in GL(4,𝐾), III. Chebyshevskii Sbornik. 2025;26(4):149-173. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-4-149-173
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