Transvections in subgroups of the general linear group containing a nonsplit maximal torus

Koibaev, V.

doi:10.1090/S1061-0022-2010-01114-2

Transvections in subgroups of the general linear group containing a nonsplit maximal torus
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by V. A. Koibaev
Translated by: P. P. Petrov

St. Petersburg Math. J. 21 (2010), 731-742

DOI: https://doi.org/10.1090/S1061-0022-2010-01114-2

Published electronically: July 14, 2010

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Abstract:

The objects of the study are intermediate subgroups of the general linear group $\mathrm {GL}(n,k)$ of degree $n$ over an arbitrary field $k$ that contain a nonsplit maximal torus associated with an extension of degree $n$ of the ground field $k$ (minisotropic torus). It is proved that if an overgroup of a nonsplit torus contains a one-dimensional transformation, then it contains an elementary transvection at some position in every column, and similarly for rows. This result makes it possible to associate net subgroups with groups of the above class and thus forms a base for their further study. This step is motivated by extremely high complexity of the lattice of intermediate subgroups. For a finite field, the lattice of overgroups of a nonsplit maximal torus is essentially determined by subfields intermediate between the ground field and its extension (G. M. Seitz, W. Kantor, R. Dye). Nothing like that holds true for an infinite field; even for the group $\mathrm {GL}(2,k)$ this lattice has much more complicated structure and essentially depends on the arithmetic of the ground field $k$ (Z. I. Borewicz, V. P. Platonov, Chan Ngoc Hoi, the author, and others).

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