To the question of the closure of the carpet

Elizaveta N. Troyanskaya

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 5, pp. 684-688

Résumé

For a root system $\Phi$, the set $\mathfrak{A}= \{\mathfrak{A}_{r}\ | \ r \in \Phi \}$ of additive subgroups $\mathfrak{A}_{r}$ over commutative ring $K$ is called a carpet of type $\Phi$ if commuting two root elements $x_{r}(t), t \in \mathfrak{A}_{r}$ and $x_{s}(u), u \in \mathfrak{A}_{s}$, gives a result where each factor lies in the subgroup $\Phi (\mathfrak{A})$ generated by the root elements $x_{r}(t), t \in \mathfrak{A}_{r}, r \in \Phi$. The subgroup $\Phi (\mathfrak{A})$ is called a carpet subgroup. It defines a new set of additive subgroups $\overline{\mathfrak{A}} = \{\overline{\mathfrak{A}}_{r} | r \in \Phi \}$, the name of the closure of the carpet $\mathfrak{A}$, which is set by equation $\overline{\mathfrak{A}}_{r} = \{t \in K\ | \ x_{r}(t) \in \Phi(\mathfrak{A})\}$. Ya. Nuzhin wrote down the following question in the Kourovka notebook. Is the closure $\overline{\mathfrak{A}}$ of a carpet $\mathfrak{A}$ {\it a carpet too? (question 19.61). The article provides a partial answer to this question. It is proved that the closure of a carpet of type $\Phi$ over commutative ring of odd characteristic $p$ is a carpet if $3$ does not divide $p$ when $\Phi$ of type $G_{2}$.

Détail

Keywords: commutative ring, Chevalley group, carpet of additive subgroups, $K$-character.