论文标题
特征零中排列组的向量不变性
Vector invariants of permutation groups in characteristic zero
论文作者
论文摘要
我们考虑一个有限的置换组自然作用于vector Space $ v $上的$ \ bbbk $。众所周知的Göbel定理断言，不变的$ \ bbbk [v]^g $的相应环是由学位的不变性生成的，最多是$ \ binom {\ dim v} {2} $。在本说明中，我们表明，如果$ \ bbbk $的特征为零，则矢量coinvariants $ \ bbbk [v^m] _g $的最高程度在上面由$ \ binom {\ binom {\ binom {\ dim v} {2} $限制，这意味着该binom dem bount $ \ binom $ \ binom v} $ \ bbbk [v^m]^g $。因此，戈贝尔的界限几乎也适用于特征零的向量不变。
We consider a finite permutation group acting naturally on a vector space $V$ over a field $\Bbbk$. A well known theorem of Göbel asserts that the corresponding ring of invariants $\Bbbk[V]^G$ is generated by invariants of degree at most $\binom{\dim V}{2}$. In this note we show that if the characteristic of $\Bbbk$ is zero then the top degree of vector coinvariants $\Bbbk[V^m]_G$ is also bounded above by $\binom{\dim V}{2}$, which implies the degree bound $\binom{\dim V}{2}+ 1$ for the ring of vector invariants $\Bbbk[V^m]^G$. So Göbel's bound almost holds for vector invariants in characteristic zero as well.
