Handbook of Teichmüller Theory, Volume II

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pp: 685–730

DOI: 10.4171/055-1/17

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

Sean Lawton[1] and Elisha Peterson[2]

(1) Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, VA 22030, FAIRFAX, UNITED STATES
(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 theory, diagrammatic calculus, SL(2,ℂ)-representations, trace functions, spin network, character variety, central functions