Minimal Affine Coordinates for SL(3,C) Character Varieties of Free Groups

TL;DR

This paper identifies minimal generators for the coordinate ring of SL(3,C) character varieties of free groups, providing explicit global coordinates and insights into their affine embeddings.

Contribution

It determines minimal generators for the coordinate ring of SL(3,C) character varieties, offering explicit coordinates and invariant theoretic relations.

Findings

Explicit minimal generators for the coordinate ring.

Global coordinate systems for the moduli.

Invariant theoretic correspondences established.

Abstract

Let X 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. 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).