论文标题
多向量表示中的基本组的溢出
Overgroups of elementary groups in polyvector representations
论文作者
论文摘要
我们在通用线性组$ gl _ {\ binom {n} {m}}}（r）$上启动了$ h $的研究，该研究包含$ m $ r $ $ r $，该$ r $ $ \ big \ bigwegge^me_n（r）$。每个这样的组$ h $都对应于唯一定义的级别$（a_0，\ dots，a_ {m-1}）$，其中$ a_0，\ dots，a_ {m-1} $是具有某些关系的$ r $的理想。在外部正方形的关键情况下，我们指出亚组晶格是标准的。换句话说，对于$ \ bigwedge^2e_n（r）$，所有中间子组$ h $均由单个理想的$ r $ $ $。此外，我们将$ \ bigwedge^mgl_n（r）$描述为不变形式系统的稳定器。该结果以代数封闭字段而经典而闻名，在这里我们证明了相应的组方案在$ \ mathbb {z} $上平滑。因此，最后的结果是在任意的交换环上。
We initiate the study of subgroups $H$ of the general linear group $GL_{\binom{n}{m}}(R)$ over a commutative ring $R$ that contain the $m$-th exterior power of an elementary group $\bigwedge^mE_n(R)$. Each such group $H$ corresponds to a uniquely defined level $(A_0,\dots,A_{m-1})$, where $A_0,\dots,A_{m-1}$ are ideals of $R$ with certain relations. In the crucial case of the exterior squares, we state the subgroup lattice to be standard. In other words, for $\bigwedge^2E_n(R)$ all intermediate subgroups $H$ are parametrized by a single ideal of the ring $R$. Moreover, we characterize $\bigwedge^mGL_n(R)$ as the stabilizer of a system of invariant forms. This result is classically known for algebraically closed fields, here we prove the corresponding group scheme to be smooth over $\mathbb{Z}$. So the last result holds over arbitrary commutative rings.
