AI 中文总结
该研究探讨混合辫群$B_{n,\mathcal{P}}$在$\mathbb{P}^1$循环分支覆盖一阶上同调的单值群作用，构造其显式张成集与埃尔米特相交形式，将其约化后与多元Burau表示建立同构关系。
AI 中文摘要
我们研究混合辫群$B_{n,\mathcal{P}}$在$\mathbb{P}^1$的循环分支覆盖的一阶上同调上的单值群作用，这些循环分支覆盖由分支点的等分歧划分相互确定。该单值群表示分解为甲板变换的$t$特征子空间上的不可约表示。对于每个特征子空间，我们使用Pochhammer围道和8字形曲线的提升构造了显式张成集，并计算了埃尔米特相交形式。该表示通过丢弃划分中$t$不可见部分（即具有平凡局部单值群的部分）而分解为约化混合辫群。在该约化表示中，当对应张成类非迷向时，每个生成元通过复反射作用；当张成类迷向时，每个生成元通过酉平移作用。若$\infty$具有非平凡局部单值群，则分解后的表示同构于在$t$处求值的约化多元Burau表示。
英文摘要
We study the monodromy action of the mixed braid group $B_{n,\mathcal{P}}$ on the first cohomology of cyclic branched covers of $\mathbb{P}^1$, which are mutually determined by a partition of branch points by equal ramification. The monodromy representation splits into irreducible representations on the $t$-eigenspaces of the deck transformation. For each, we construct an explicit spanning set using lifts of Pochhammer contours and figure-eight curves, and compute the Hermitian intersection form. The representation factors through a reduced mixed braid group by dropping $t$-invisible parts of the partition (those with trivial local monodromy). In this reduced representation, each generator acts by a complex reflection when the corresponding spanning class is non-isotropic, and by a unitary transvection when it is isotropic. Provided $\infty$ has non-trivial local monodromy, the factored representation is isomorphic to the reduced multivariate Burau representation evaluated at $t$.
Comments38 pages, 19 figures
