关于特征2中的正交不变量
On orthogonal invariants in characteristic 2
摘要：
研究了特征为2的代数闭基域$k$上的不变量环$R^G$，其中$G$是正交群O（n）或特殊正交群SO（n），自然作用在标准向量表示的$m$ -倍直和$k^n \oplus...\oplus k^n$的坐标环$R$上.证明了对于O（2），$O(3)=SO(3)$，SO（4）和O（4），存在一个$m$ -线性不变量，其中$m$为任意大，它不能表示为低次不变量的多项式.这是在鲜明的对比与统一的描述环的不变量有效的所有其他特点，并支持的猜想，同样的现象发生在所有$n$。对于一般的偶$n$，构造了新的O（n）-不变量，这些不变量不能表示为二次不变量的多项式.与这些结果相比，它表明，有理不变量有一个统一的描述有效的所有特征。类似地，如果是$m \leq n$，则$R^O(n)$由明显的不变量生成。对于所有的$n$，代数$R^G$是由二次不变量生成的子代数上的一个二次生成模，对于奇数的$n$，任何SO（n）-不变量的平方是二次不变量的多项式。最后我们提到，对于偶数$n$，给出了一个$n$ -线性SO（n）-不变量，它区分了SO（n）和O（n）（就像行列式在所有不同于2的特征中一样）.
英文摘要：
Working over an algebraically closed base field $k$ of characteristic 2, the ring of invariants $R^G$ is studied, where $G$ is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring $R$ of the $m$-fold direct sum $k^n \oplus...\oplus k^n$ of the standard vector representation. It is proved for O(2), $O(3)=SO(3)$, SO(4), and O(4), that there exists an $m$-linear invariant with $m$ arbitrarily large, which is not expressible as a polynomial of invariants of lower degree. This is in sharp contrast with the uniform description of the ring of invariants valid in all other characteristics, and supports the conjecture that the same phenomena occur for all $n$. For general even $n$, new O(n)-invariants are constructed, which are not expressible as polynomials of the quadratic invariants. In contrast with these results, it is shown that rational invariants have a uniform description valid in all characteristics. Similarly, if $m \leq n$, then $R^O(n)$ is generated by the obvious invariants. For all $n$, the algebra $R^G$ is a finitely generated module over the subalgebra generated by the quadratic invariants, and for odd $n$, the square of any SO(n)-invariant is a polynomial of the quadratic invariants. Finally we mention that for even $n$, an $n$-linear SO(n)-invariant is given, which distinguishes between SO(n) and O(n) (just like the determinant in all characteristics different from 2).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
