Minimal affine coordinates for [formula omitted] character varieties of free groups

Abstract

Let X r be the moduli of SL ( 3 , C ) representations of a rank r free group. In this paper we determine minimal generators of the coordinate ring of X r . This at once gives explicit global coordinates for the moduli and determines the dimension of the moduli's minimal affine embedding. Along the way, we utilize results concerning the moduli of r-tuples of matrices in gl ( 3 , C ) . Consequently, we also state general invariant theoretic correspondences between the coordinate rings of the moduli of r-tuples of elements in gl ( n , C ) , sl ( n , C ) , and SL ( n , C ) .