郑文化,南基洙.T-函子与超曲面[J].数学研究及应用,2018,38(1):85~94
T-函子与超曲面
Lanne's \textbf{T}-functor and Hypersurfaces
投稿时间:2016-09-17  修订日期:2017-09-01
DOI:10.3770/j.issn:2095-2651.2018.01.008
中文关键词:  \textbf{T}-函子  超曲面  稳定子群
英文关键词:\textbf{T}-functor  hypersurface  pointwise stabilizers
基金项目:国家自然科学基金资助 (Grant No.11371343).
作者单位
郑文化 大连理工大学数学科学学院, 辽宁 大连 116024 
南基洙 大连理工大学数学科学学院, 辽宁 大连 116024 
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中文摘要:
      本文通过讨论\textbf{T}-函子在不变理想上的作用,首先证明了\textbf{T}-函子在不稳定代数范畴中只能降低嵌入维数,并推导出\textbf{T}-functor在不稳定代数范畴中保持超曲面的结论. 把上面结果应用到不变式理论,我们用新方法证明了下面的结论:当一个有限群的不变式为超曲面时,它稳定子群的不变式仍然是超曲面. 最后文章通过几个反例指出当一个群的稳定子群或者Sylow $p$-子群的不变式为超曲面时,该群本身的不变式不一定是超曲面.
英文摘要:
      Through discussing the transformation of the invariant ideals, we firstly prove that the \textbf{T}-functor can only decrease the embedding dimension in the category of unstable algebras over the Steenrod algebra. As a corollary we obtain that the \textbf{T}-functor preserves the hypersurfaces in the category of unstable algebras. Then with the applications of these results to invariant theory, we provide an alternative proof that if the invariant of a finite group is a hypersurface, then so are its stabilizer subgroups. Moreover, by several counter-examples we demonstrate that if the invariants of the stabilizer subgroups or Sylow $p$-subgroups are hypersurfaces, the invariant of the group itself is not necessarily a hypersurface.
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