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Two-dimensional representations of the free group in two generators over an arbitrary field. (English) Zbl 1007.20021

A classification of two-dimensional complex representations of the free group with two generators was given in the book by K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida [From Gauß to Painlevé. A modern theory of special functions, Braunschweig, Vieweg (1991; Zbl 0743.34014)]. In the present paper the authors extend this representation in the case of an arbitrary field. Let \(G=(u_1,u_2)\) be the free group on two generators, \(V\) a two dimensional vector space over an arbitrary field \(F\) and \(\rho\colon G\to\text{GL}(V)\) a two-dimensional representation. If \(g_i=\rho(u_i)\), \(i=1,2\), \(g_3=\rho(u_1u_2)^{-1}\) and \(t_i=\text{tr}(g_i)\), \(i=1,2,3\), \(e_i=\det(g_i)\), \(i=1,2,3\), then the authors give the classification by describing all possible 5-tuples \((t_1,t_2,t_3,e_1,e_2)\) in \(F\). In their second theorem they deal with the uniqueness of the representation for a 5-tuple \((t_1,t_2,t_3,e_1,e_2)\).

MSC:

20E05 Free nonabelian groups
20C15 Ordinary representations and characters

Citations:

Zbl 0743.34014
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References:

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