Spin networks and SL(2,C)-Character varieties

TL;DR

This paper constructs a canonical basis for the regular functions on the SL(2,C)-character variety of a free group on two generators using spin networks, providing new insights into its structure and symmetries.

Contribution

It introduces a novel basis for the coordinate ring of the SL(2,C)-character variety of F_2 using spin networks and graphical calculus, offering a new proof of classical results.

Findings

Established an isomorphism between coordinate rings and matrix coefficients.

Determined the symmetries and multiplicative structure of the basis.

Provided a canonical description of regular functions on the character variety.

Abstract

Denote the free group on 2 letters by F_2 and the SL(2,C)-representation variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]^SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the SL(2,C)-character variety of F_2 and a new proof of a classical result of Fricke, Klein, and Vogt.