非可定向表示球面
Non-orientable representation spheres
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中文总结 AI 辅助
该研究针对有限群提出不可约实表示对应的表示球面非可定向性问题，证明具正规Sylow 2-子群的群答案为是，给出阶112的2-幂零群反例，关联Burnside环单位群并提出相关猜想。
中文摘要 AI 辅助
对于任意有限群，我们提出如下问题：给定一个不可约实表示，当且仅当该表示为实型非平凡表示时，对应的表示球面是否是非可定向的？我们证明，若群具有正规Sylow 2-子群，则该问题的答案为是；但通过初等论证，我们给出一个阶为112的2-幂零群，其答案为否。我们还将该问题与Burnside环的单位群关联，其中我们恢复了Bouc为2-群发现的一个基，并提出了一个关于从可解子群检测非可定向性的猜想。该问题的动机源于有限群的置换扭曲上同调构造中出现的问题。
英文摘要
For any finite group, we pose the following question: given an irreducible real representation, is the associated representation sphere non-orientable if and only if the representation is nontrivial of real type? We prove that the answer to this question is yes if the group has a normal Sylow 2-subgroup, but exhibit a 2-nilpotent group of order 112 for which the answer is no via elementary arguments. We also link the question to the unit group of the Burnside ring, where we recover a basis discovered by Bouc for 2-groups, and pose a conjecture about detection of non-orientability from solvable subgroups. The question is motivated by issues arising from the construction of permutation twisted cohomology for finite groups.
