Transvection modules in the overgroups of a~non-split maximal torus

N. A. Dzhusoeva ; V. A. Koibaev

Vladikavkazskij matematičeskij žurnal, Tome 16 (2014) no. 3, pp. 3-8

Résumé

The aim of this article is to investigate the modules of transvections and rings of multipliers subgroups of the general linear group $G=GL(n,k)$ of degree $n$ over a field $k$, containing non-split maximal torus $T=T(d)$, associated with a radical extension of $k(\sqrt[n]d)$ of the degree $n$ of the ground field $k$ of an odd characteristic (minisotropic torus). We find a full list of $2\cdot[(\frac{n-1}2)^2]$ relations ($[\cdot]$ – integer part) of the modules of transvections. We prove that all ring of multipliers coincide, and all modules transvections are ideals of the ring of multipliers. All results were proved by the assumption that the ground field $k$ is the field of fractions of a principal ideal domain.