Abstract:
In the present paper we give a new proof of the description of irreducible subgroups generated by root subgroups in special linear group $\mathrm{SL}(n,K)$. For the first time such a description was appeared in J. McLaughlin's work. His work was one of the first papers dedicated to study of the generations by long root unipotent subgroups in Chevalley groups. At the present time the geometry of long root subgroups is a well-established field. But it remains a lot of unsolved tasks about short root unipotent subgroups. In particular, the description of irreducible subgroups generated by short root subgroups in exceptional Chevalley groups over arbitrary field is unknown. In our proof we consider the group $\mathrm{SL}(n,K)$ as the Chevalley group of type $\mathscr{A}_\ell$. Thus in our opinion it is possible to use the given approach for study of the irreducible subgroups generated by short root subgroups in Chevalley groups.
Keywords:general linear group, special linear group, unipotent root subgroups, irreducible subgroups.
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