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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 {\it 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:
20E05Free nonabelian groups
20C15Ordinary representations and characters of groups
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References:
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