We compute the invariants of Weyl groups of type $A_n$, $B_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$ in mod 2 Milnor $K$-theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order and where $-1$ is a square, the invariants decompose as direct sums of the coefficient module. With the exception of invariants coming from $G_2$-components, all basis elements are induced either by Stiefel-Whitney classes or specific invariants in the Witt ring. The proof is based on Serre's splitting principle that guarantees detection of invariants on elementary abelian 2-subgroups generated by reflection.

中文翻译：

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关于Weyl群的mod 2上同调不变量的可分解性

我们在 mod 2 Milnor $K$-theory 中计算 $A_n$、$B_n$、$D_n$、$E_6$、$E_7$、$E_8$、$F_4$ 和 $G_2$ 类型的 Weyl 群的不变量和更一般的循环模块，它们被 2 湮灭。 在群阶特征互质的基域上，其中 $-1$ 是平方，不变量分解为系数模块的直接和。除了来自 $G_2$-components 的不变量之外，所有基元素都由 Stiefel-Whitney 类或 Witt 环中的特定不变量引起。该证明基于 Serre 的分裂原理，该原理保证检测到由反射生成的基本阿贝尔 2-子群上的不变量。