模不变理论中韦劳范数猜想的反例
Counterexamples to Norm Conjectures of Wehlau in Modular Invariant Theory
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中文总结 AI 辅助
研究有限群\(G\)及有限维\(G -\)模\(V\)在特征\(2\)下韦劳范数猜想，通过构造\(C_2^3\)和\(C_2^4\)在\(\mathbb F_8\)上的忠实四维表示给出反例，反驳了韦劳范数猜想及其无限制扩展。
中文摘要 AI 辅助
设\(G\)为有限群，\(V\)为域\(k\)上的有限维\(G\)-模。我们在特征\(2\)下构造了关于韦劳范数猜想的明确反例。给出了\(C_2^3\)和\(C_2^4\)在\(\mathbb F_8\)上的忠实四维表示，其中每个非线性轨道范数可分解；在\(C_2^4\)的例子中，不变环是多项式，从而反驳了韦劳的范数猜想及其无限制扩展。
英文摘要
Let $G$ be a finite group and let $V$ be a finite-dimensional $G$-module over a field $k$. We construct explicit counterexamples in characteristic $2$ to conjectures of Wehlau concerning norms. We give faithful four-dimensional representations of $C_2^3$ and $C_2^4$ over $\mathbb F_8$ for which every nonlinear orbit norm is decomposable; in the $C_2^4$ example, the invariant ring is polynomial, thereby disproving both Wehlau's norm conjecture and its unrestricted extension.
