Invariants under simultaneous conjugation of SL(2) matrices

Given tuples of SL(2)-matrices, one can look at which functions in their coordinate ring do not change when we simultaneously conjugate the matrices: these are called the invariant functions. Our interest in them is motivated by the fact that these tuples, up to simultaneous conjugation, occur as the so called "monodromy group" of certain linear differential equations. We will look at these invariant functions from three different perspectives. First, we employ classical invariant theory to find the structure of the space generated by these invariant functions. Next, we use geometric invariant theory to define a "quotient" using these invariant functions. Finally, we place the results in the more general setting of representation theory by looking at the structure of the space of matrices as a SL(2)-representation.