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Subgroups of Chevalley groups of types 𝐵_{𝑙} and 𝐶_{𝑙} containing the group over a subring, and corresponding carpets
St. Petersburg Mathematical Journal ( IF 0.7 ) Pub Date : 2020-06-11 , DOI: 10.1090/spmj/1620 Ya. N. Nuzhin , A. V. Stepanov
St. Petersburg Mathematical Journal ( IF 0.7 ) Pub Date : 2020-06-11 , DOI: 10.1090/spmj/1620 Ya. N. Nuzhin , A. V. Stepanov
Abstract:This is a continuation of the study of subgroups of the Chevalley group 
over a ring 
with root system 
and weight lattice 
that contain the elementary subgroup 
over a subring 
of 
. A. Bak and A. V. Stepanov considered recently the case of the symplectic group (simply connected group with root system 
) in characteristic 2. In the current article, that result is extended to the case of 
and for the groups with other weight lattices. Like in the Ya. N. Nuzhin's work on the case where 
is an algebraic extension of a nonperfect field 
and 
is not simply laced, the description involves carpet subgroups parametrized by two additive subgroups. In the second part of the article, the Bruhat decomposition is established for these carpet subgroups and it is proved that they have a split saturated Tits system. As a corollary, it is shown that they are simple as abstract groups.
中文翻译：
类型为𝑙_{and}和𝐶_{𝑙}的Chevalley分组的子分组，在子环上包含该分组，并带有相应的地毯
摘要：这是对Chevalley群子群在具有根系统和权重格的环上的研究的延续，该环具有的子环上的基本子群。A. Bak和A. V. Stepanov最近在特征2中考虑了辛群（具有根系统的简单连通群）的情况。在当前文章中，该结果扩展到以及具有其他权重格的群的情况。像在雅。N. Nuzhin在非理想场的代数扩展和 不仅是简单的绑带，其描述还涉及由两个附加子组参数化的地毯子组。在本文的第二部分中，为这些地毯亚组建立了Bruhat分解，并证明了它们具有分裂的饱和Tits系统。结果表明，它们作为抽象组很简单。
更新日期：2020-06-11
over a ring with root system and weight lattice that contain the elementary subgroup over a subring of . A. Bak and A. V. Stepanov considered recently the case of the symplectic group (simply connected group with root system ) in characteristic 2. In the current article, that result is extended to the case of and for the groups with other weight lattices. Like in the Ya. N. Nuzhin's work on the case where is an algebraic extension of a nonperfect field and is not simply laced, the description involves carpet subgroups parametrized by two additive subgroups. In the second part of the article, the Bruhat decomposition is established for these carpet subgroups and it is proved that they have a split saturated Tits system. As a corollary, it is shown that they are simple as abstract groups. 中文翻译：
类型为𝑙_{and}和𝐶_{𝑙}的Chevalley分组的子分组，在子环上包含该分组，并带有相应的地毯
摘要：这是对Chevalley群子群在具有根系统和权重格的环上的研究的延续，该环具有的子环上的基本子群。A. Bak和A. V. Stepanov最近在特征2中考虑了辛群（具有根系统的简单连通群）的情况。在当前文章中，该结果扩展到以及具有其他权重格的群的情况。像在雅。N. Nuzhin在非理想场的代数扩展和
不仅是简单的绑带，其描述还涉及由两个附加子组参数化的地毯子组。在本文的第二部分中，为这些地毯亚组建立了Bruhat分解，并证明了它们具有分裂的饱和Tits系统。结果表明，它们作为抽象组很简单。
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