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Invariants of 2 by 2 matrices, irreducible SL(2,C) characters and the Magnus trace map

Carlos A. A. Florentino
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Abstract

We obtain an explicit characterization of the stable points of the action of G=SL(2,C) on the cartesian product G^n by simultaneous conjugation on each factor, in terms of the corresponding invariant functions, and derive from it a simple criterion for irreducibility of representations of finitely generated groups into G. We also obtain analogous results for the action of SL(2,C) on the vector space of n-tuples of 2 by 2 complex matrices. For a free group F_n of rank n, we show how to generically reconstruct the 2^{n-2} conjugacy classes of representations F_n -> G from their values under the map T_n : G^n = Hom(F_n,G) -> C^{3n-3} considered in [M], defined by certain 3n-3 traces of words of length one and two.

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Carlos A. A. Florentino. 2006-03-02. Invariants of 2 by 2 matrices, irreducible SL(2,C) characters and the Magnus trace map. https://doi.org/10.1007/s10711-006-9097-5

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