Handbook of Teichmüller Theory, Volume II
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(2) Department of Mathematical Sciences, United States Military Academy, NY 10996, West Point, UNITED STATES
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.
Keywords: Invariant thaory, diagrammatic calculus, SL(2,ℂ)-representations, trace functions, spin network, character variety, central functions