TY - JOUR

T1 - Subring subgroups in symplectic group in characteristic 2

AU - Bak, A.

AU - Stepanov, A.

N1 - A. Bak, A. Stepanov, “Subring subgroups of symplectic groups in characteristic 2”, Алгебра и анализ, 28:4 (2016), 47–61; St. Petersburg Math. J., 28:4 (2017), 465–475

PY - 2016

Y1 - 2016

N2 - In 2012 the second author obtained a description of the lattice of subgroups of a Chevalley group $G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided $\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$. It turns out that this lattice is a disjoint union of ``sandwiches'' parameterized by subrings $R$ such that $K\subseteq R\subseteq A$. In the current article a similar result is proved for $\Phi=C_n$, $n\ge3$, and $2=0$ in $K$. In this setting one has to introduce more sandwiches, namely, sandwiches which are parameterized by form rings $(R,\Lambda)$ such that $K\subseteq\Lambda\subseteq R\subseteq A$. The result we get generalizes Ya.\,N.\,Nuzhin's theorem of 2013 concerning the root systems $\Phi=B_n,$ $C_n$, $n\ge3$, where the same description of the subgroup lattice is obtained, but under the condition that $A$ is an algebraic extension of a field~$K$.

AB - In 2012 the second author obtained a description of the lattice of subgroups of a Chevalley group $G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided $\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$. It turns out that this lattice is a disjoint union of ``sandwiches'' parameterized by subrings $R$ such that $K\subseteq R\subseteq A$. In the current article a similar result is proved for $\Phi=C_n$, $n\ge3$, and $2=0$ in $K$. In this setting one has to introduce more sandwiches, namely, sandwiches which are parameterized by form rings $(R,\Lambda)$ such that $K\subseteq\Lambda\subseteq R\subseteq A$. The result we get generalizes Ya.\,N.\,Nuzhin's theorem of 2013 concerning the root systems $\Phi=B_n,$ $C_n$, $n\ge3$, where the same description of the subgroup lattice is obtained, but under the condition that $A$ is an algebraic extension of a field~$K$.

KW - symplectic group

KW - commutative ring

KW - subgroup lattice

KW - Bak unitary group

KW - group identity with constants

KW - small unipotent element

KW - nilpotent structure of K_1

UR - https://elibrary.ru/item.asp?id=26414194

M3 - Article

VL - 28

SP - 47

EP - 61

JO - АЛГЕБРА И АНАЛИЗ

JF - АЛГЕБРА И АНАЛИЗ

SN - 0234-0852

IS - 4

ER -