On irreducible carpets of additive subgroups of type $G_2$ over fields of characteristic $p>0$

This article is devoted to the study of subgroups of Chevalley groups defined by carpets, 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 it defines, does not contain new root elements. One of the important questions in the study of carpet subgroups is the question of the closeness of the original carpet. In is known that this question is reduced to irreducible carpets, that is, to carpets all additive subgroups of which are nonzero [1, 2]. The article describes irreducible carpets of type $G_2$ over a field $K$ of characteristic $p>0$, at least one additive subgroup of which is an $R$-module, in the case when $K$ is an algebraic extension of the field $R$.