TITLE: explicit linear representations of fundamental groups of surfaces
QUESTION [5 upvotes]: I am looking for an explicit representation of the fundamental group of a closed orientable surface of genus >1. I guess they should be abundant in degree 2. Did anyone see the explicit matrix construction of such a representation? Are there any integral ones? Maybe in higher degrees?

REPLY [2 votes]: Here are the examples from Narasimhan and Seshadri that I mentioned in my comment above.  (I could have the reference wrong; I'm actually taking this out of notes from a talk I gave many years ago.)
These examples are really not that exciting, but at least they give irreducible unitary representations of each degree.  Writing the generators of $\pi_1 M^g$ as $a_i$ and $b_i$ ($i=1, \ldots, g$) we define a representation $\rho$ by sending $a_1$ to the diagonal matrix
$$A_1 = \left[\begin{array}{rrr} z_1 \\ & \ddots\\ && z_n \end{array}\right],$$ 
where the $z_i$ are distinct, and sending $a_2$ to the permutation matrix 
$$A_2 = \left[\begin{array}{rrrrrr} 0 & 0 &  \cdots & 0 &1 \\ 1 & 0  & \cdots && 0 \\
0 & \ddots & 0 & \cdots & 0\\
 0 & \cdots &   1&      0 & 0\\
  0 & \cdots &  0 &      1 & 0\end{array}\right].$$  All the other generators are sent to the identity matrix.  Since the $b_i$ all go to the identity, all the commutators $[\rho(a_i), \rho(b_i)]$ are trivial and we have a representation.  It's irreducible because the invariant subspaces of $A_1$ are just sums of eigenspaces, and these eigenspaces are permuted transitively by $A_2$.  
One can vary this example in a number of ways to get other interesting examples.