Minimal affine coordinates for character varieties of free groups

Let be the moduli of representations of a rank r free group. In this paper we determine minimal generators of the coordinate ring of . This at once gives explicit global coordinates for the moduli and determines the dimension of the moduliʹs minimal affine embedding. Along the way, we utilize results concerning the moduli of r-tuples of matrices in . Consequently, we also state general invariant theoretic correspondences between the coordinate rings of the moduli of r-tuples of elements in , , and .