Let $Y_{n}$ be what I'm calling the ordered configuration space, the topological space of all ordered subsets of $\mathbb{C}$ of cardinality $n$. This can be viewed as $$\mathrm{Spec}(\mathbb{C}[t_{1}, ... , t_{n}, \{(t_{i} - t_{j})^{-1}\}_{1 \leq i < j \leq n}]).$$

The (topological) fundamental group of this is the pure braid group $P_{n}$. So by a general version of the Riemann Existence Theorem, the elements of $P_{n}$ must be Galois automorphisms of the maximal Galois extension of the function field $\mathbb{C}(t_{1}, ... , t_{n})$ which is unramified except at the primes $(t_{i} - t_{j})$. How may I describe algebraically the Galois automorphisms corresponding to, say, the generators of $P_{n}$?

I have a guess (something along the lines of each generator $A_{i,j}$ of $P_{n}$ sending $(t_{i} - t_{j})^{1/n} \mapsto \zeta_{n}(t_{i} - t_{j})^{1/n}$, except that seems to make the Galois group abelian) but am not sure how to prove anything systematically, and I can't find this explicitly described in any source. Thank you for any help you can give.