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

Abstract

Denote the free group on 2 letters by F2 and the SL(2,ℂ)-representation variety of F2 by R = Hom(F2,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 F2 and a new proof of a classical result of Fricke, Klein, and Vogt.