Arrangement of the subgroups that contain an unramified quadratic torus in the general linear group of degree 2 over a local number field (p≠2)

Бондаренко, А. А.

doi:10.1007/bf02434846

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“…As an example, we recall what the ring R 0 looks like in the case of GL(2, k); see [1]- [4], [9]- [13], [16]. If k is a local number field (a finite extension of the field of p-adic numbers, p = 2), and if the torus is determined by a nonramified quadratic extension of k, then the ring R 0 coincides with the ring of integers of the normalized valuation of k; in this case, there are countably many intermediate subgroups.…”

Section: §1 Introductionmentioning

confidence: 99%

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

Koibaev

2010

St. Petersburg Math. J.

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Abstract. The objects of the study are intermediate subgroups of the general linear group 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 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). §1. Introduction Through the last decades, a lot of attention has been paid to the study of subgroups of classical and Chevalley groups that contain a split maximal torus. For the case of a split maximal torus, there is an almost complete description of the lattice of its overgroups in classical groups, and in Chevalley groups. The overgroups of split maximal tori were studied by A. Borel, J. Tits, G. M. Seitz, O. King, and others. We mention the important contribution of the Leningrad and St.-Petersburg algebra school (Z. I. Borewicz, N. A. Vavilov, and their students) to the description of overgroups of diagonal and block diagonal subgroups, over fields and rings. In the surveys [5, 6, 7, 8] and [27] one can find detailed descriptions of results in this direction.Even for the general linear group over a field, much less is known about the description of overgroups of nonsplit maximal tori associated with a ground field extension (minisotropic tori). In a more general setting, this problem (related to the description of overgroups of subgroups from the Aschbacher class C 3 ) can be stated as follows. Let K/k be a finite extension of degree m, let V be a vector space of dimension n over the field K (and thus, of dimension mn over k). Then, obviously (a K-linear map is k-linear), we have GL K (V ) ≤ GL k (V ), or, in the matrix form, GL(n, K) ≤ GL(mn, k). Observe that, for n = 1, the group GL(1, K) = K * is a nonsplit maximal torus. We state a result due to Shang Zhi Li [23], which reduces the above problem to the special case of a nonsplit torus.

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Section: §1 Introductionmentioning

confidence: 99%

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

Koibaev

2010

St. Petersburg Math. J.

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Arrangement of the subgroups that contain an unramified quadratic torus in the general linear group of degree 2 over a local number field (p≠2)

Cited by 1 publication

References 1 publication

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

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

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