On irreducible carpets of additive subgroups of type $G_2$

This article discusses the subgroups of Chevalley groups, defined by carpets — the sets of additive subgroups of the main definition ring. Such subgroups are called carpet subgroups and they are generated by root elements with coefficients from the corresponding additive subgroups. By definition, a carpet is closed if the carpet subgroup, which it defines, does not contain new root elements. One of the fundamentally important issues in the study of carpet subgroups is the problem of the closure of the original carpet. It is known that this problem is reduced to irreducible carpets, that is, to carpets, all additive subgroups of which are nonzero [8; 11]. This paper describes irreducible carpets of type $G_2$ over a field $K$ of characteristics $p>0$, all additive subgroups of which are $R$-modules, in case when $K$ is an algebraic extension of $R$. It is proved that such carpets are closed and can be parametrized by two different fields only for $p=3$, and for other $p$ they are determined by one field. In this case the corresponding carpet subgroups coincide with Chevalley groups of type $G_2$ over intermediate subfields $P$, $R \subseteq P \subseteq K$.