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Geometry of 1-tori in \(\text{GL}_n\). (English. Russian original) Zbl 1202.20054

St. Petersbg. Math. J. 19, No. 3, 407-429 (2008); translation from Algebra Anal. 19, No. 3, 119-150 (2007).
Summary: We describe the orbits of the general linear group \(\text{GL}(n,T)\) over a skew field \(T\) acting by simultaneous conjugation on pairs of 1-tori, i.e., subgroups conjugate to \(\text{diag}(T^*,1,\dots,1)\), and identify the corresponding spans. We also provide some applications of these results to the description of intermediate subgroups and generation. These results were partly superseded by A. Cohen, H. Cuypers, and H. Sterk, but our proofs use only elementary matrix techniques. As another application of our methods, we enumerate the orbits of \(\text{GL}(n,T)\) on pairs of a 1-torus and a root subgroup, and identify the corresponding spans. This paper constitutes an elementary invitation to a series of much more technical works by the author and V. Nesterov, where similar results are established for microweight tori in Chevalley groups over a field.

MSC:

20H25 Other matrix groups over rings
20G15 Linear algebraic groups over arbitrary fields
14L35 Classical groups (algebro-geometric aspects)
20F05 Generators, relations, and presentations of groups
20G35 Linear algebraic groups over adèles and other rings and schemes
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References:

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