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We obtain an explicit characterization of the stable points of the action of  on the cartesian product   by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into  is derived. We also obtain analogous results for the action of  on the vector space of -tuples of 2 × 2 complex matrices. For a free group   of rank , we show how to generically reconstruct the 2 conjugacy classes of representations   →   from their values under the map  considered in Magnus [Math. Zeit. , 91–103 (1980)], defined by certain 3 − 3 traces of words of length one and two.

Invariants of 2 × 2 matrices, irreducible $${\hbox{SL}(2, \mathbb{C})}$$ characters and the Magnus trace map