TITLE: Explicit Riemann Hilbert correspondence
QUESTION [6 upvotes]: For simplicity, we assume that $X=\mathbb P_{\mathbb C}^1-\{s_1, s_2, \dots, s_k\}$ and $\infty \in X$.
Consider the trivial bundle $E=\mathcal O_X^r$ with the connection $\nabla$ induced by a Fushcian system on $X$, i.e. 
$$\nabla:= d+\sum_{i=1}^k \frac{A_i}{z-s_i},$$
where $A_i$'s are $r\times r$ constant matrices over $\mathbb C$ such that $\sum_{i=1}^kA_i=0$.
Then we have a flat bundle with connection (or a logarithmic connection), note it only has regular singularities.
My question is that what is the corresponding monodromy representation to $(E,\nabla)$ via the Riemann Hilbert correspondence?
The known part is that locally at each $z=s_i$, the monodromy $T_i$ can be obtained by the residue map: $T_i=\exp(2\pi i A_i)$, but this seems not to admit a unique representation up to isomorphism. 
Is there any more properties/facts around the question?

REPLY [2 votes]: You are correct that the monodromy representation is given by the $T_i$. To address your concerns about this being unique up to isomorphism, notice that a change of basis of $\mathcal{O}_X^r$ induces a corresponding (compatible) change to the $T_i$.
You might be interested in the following references: Sections 5.1.1 and 5.1.2 of Hotta, Tanasaki, and Takeuchi's "D-Modules, Perverse Sheaves, and Representation Theory"; and Chapter III of "Algebraic D-Modules" by Borel, et al.