论文标题
部分可观测时空混沌系统的无模型预测
When are the natural embeddings of classical invariant rings pure?
论文作者
论文摘要
考虑一个还原线性代数组$ g $在无限字段上在多项式环$ s $上线性作用；关键示例是通用线性群，符号群，正交组和特殊线性群，带有经典表示，如韦尔的书中：对于一般线性群，请考虑直接的标准表示和副本的副本和偶的副本；在其他情况下，采用标准表示的副本。在各个情况下，不变的环是决定性环，由矩阵交替的pfaffians定义的环，对称的确定性环和司羊grassmanians的plücker坐标环；这些是标题的经典不变环，$ s^g \ subseteq s $是天然嵌入。 在一个特征性零的领域，还原组是线性还原性的，因此，不变的环$ s^g $是$ s $的纯subring，同等地，$ s^g $是$ s $的直接汇总为$ s $，作为$ s^g $ -module。在积极特征的领域，还原群通常不再是线性还原性的。在积极的特征情况下，我们确定当包含$ s^g \ subseteq s $纯净时。事实证明，如果$ s^g \ subseteq s $是纯净的，那么不变的环$ s^g $是常规的，或者$ g $是线性还原的。
Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of $S$, equivalently, $S^G$ is a direct summand of $S$ as an $S^G$-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular, or the group $G$ is linearly reductive.
