Subgroups Generated by a Pair of 2-Tori in \(\operatorname{GL}(4,K)\), II

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/t9254-6010-7867-w Subgroups Generated by a Pair of 2-Tori in \(\operatorname{GL}(4,K)\), II Nesterov, V. V. , Zhang, M. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 3. Abstract: The present paper is the next in a large series of works devoted to the geometry of microweight tori in the Chevalley groups. Namely, we describe the subgroups generated by a pair of 2-tori in \(\operatorname{GL}(4,K)\). Recall that 2-tori in \(\operatorname{GL}(n,K)\) are the subgroups conjugate to the diagonal subgroup of the following form \(\operatorname{diag}(\varepsilon, \varepsilon, 1,\dots, 1)\). In one of the previous work we proved the reduction theorem for the pairs of \(m\)-tori. It follows that any pair of 2-tori can be embedded in \(\operatorname{GL}(6,K)\) by simultaneous conjugation. The orbit of a pair of 2-tori \((X,Y)\) is called the orbit in \(\operatorname{GL}(n,K)\), if the pair \((X,Y)\) is embedded in \(\operatorname{GL}(n,K)\) by simultaneous conjugation and it can not be embedded in \(\operatorname{GL}(n-1,K)\). Here \(n\) can take values 3, 4, 5 and 6. The most difficult and general case is the case of \(\operatorname{GL}(4,K)\). In the article we describe spans in \(\operatorname{GL}(4,K)\), corresponding to degenerate orbits. Keywords: general linear group, unipotent root subgroups, semisimple root subgroups, \(m\)-tori, diagonal subgroup Language: English Download the full text Download JATS XML For citation: Nesterov, V. V. and Zhang, M.Subgroups Generated by a Pair of 2-Tori in \(\operatorname{GL}(4,K)\), II, Vladikavkaz Math. J., 2025, vol. 27, no. 3, pp. 101-119. DOI 10.46698/t9254-6010-7867-w + References 1. Vavilov, N. A. and Nesterov, V. V. Geometry of Microweight Tori, Vladikavkaz Mathematical Journal, 2008, vol. 10, no. 1, pp. 10-23 (in Russian). 2. Nesterov, V. V. and Vavilov, N. A. Pairs of Microweight Tori in \(\operatorname{GL_n}\), Chebyshevskii Sbornik, 2020, vol. 21, no. 4, pp. 152-161. DOI: 10.22405/2226-8383-2020-21-4-152-161. 3. Vavilov, N. A. Geometry of \(1\)-Tori in \(\operatorname{GL_n}\), St. Petersburg Mathematical Journal, 2008, vol. 19, no. 3, pp. 407-429. DOI: 10.1090/S1061-0022-08-01004-2. 4. Vavilov, N. A. and Nesterov, V. V. Subgroups Generated by a Pair of \(2\)-Tori in \(\operatorname{GL(5,K)}\), Zapiski nauchnyh seminarov POMI [Notes of Scientific Seminars of POMI], 2023, vol. 522, pp. 8-45 (in Russian). 5. Nesterov, V. and Zhang, M. Subgroups Generated by a Pair of \(2\)-Tori in \(\operatorname{GL(4,K)}\). I, Zapiski nauchnyh seminarov POMI [Notes of Scientific Seminars of POMI], 2024, vol. 531, pp. 127-146. 6. Vavilov, N. A. Subgroups of Chevalley Groups Containing a Maximal Torus, Trudy Leningradskogo matematicheskogo obshchestva [Proceedings of the Leningrad Mathematical Society], 1990, vol. 1, pp. 64-109 (in Russian). 7. Vavilov, N. A. Weight Elements of Chevalley Groups, St. Petersburg Mathematical Journal, 2009, vol. 20, no. 1, p. 23-57. DOI: 10.1090/S1061-0022-08-01036-4. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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