Geometry of 1-tori in 𝐺𝐿_{𝑛}

Vavilov, N.

doi:10.1090/S1061-0022-08-01004-2

Geometry of 1-tori in $\mathrm {GL}_n$
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by N. A. Vavilov
Translated by: the author

St. Petersburg Math. J. 19 (2008), 407-429

DOI: https://doi.org/10.1090/S1061-0022-08-01004-2

Published electronically: March 21, 2008

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Abstract:

We describe the orbits of the general linear group $\operatorname {GL}(n,T)$ over a skew field $T$ acting by simultaneous conjugation on pairs of $1$-tori, i.e., subgroups conjugate to $\operatorname {diag}(T^*,1,\ldots ,1)$, 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 These results were partly superseded by A. Cohen, H. Cuypers, and H. Sterk, but our proofs use only elementary matrix techniques.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 $\operatorname {GL}(n,T)$ 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.

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