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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 $(t_1,t_2,t_3,e_1,e_2)$.

20E05Free nonabelian groups
20C15Ordinary representations and characters of groups
Full Text: DOI
[1] Boularas, D.; Bouzar, Z.: Concomitants et p-uplets de matrices $2{\times}$2. Linear and multilinear algebra 41, 161-173 (1996) · Zbl 0869.15020
[2] Formanek, E.; Halpin, P.; Li, W.: The Poincaré series of the ring of $2{\times}2$ generic matrices. J. algebra 69, 105-112 (1981) · Zbl 0459.16013
[3] E. Formanek, The invariants of n{$\times$}n matrices, in: Invariant Theory, Lecture Notes in Mathematics, vol. 1278, Springer, Berlin, 1987, pp. 18--43 · Zbl 0645.16012
[4] Friedland, S.: Simultaneous similarity of matrices. Adv. in math. 50, 189-265 (1983) · Zbl 0532.15009
[5] Iwasaki, K.; Kimura, H.; Shimomura, S.; Yoshida, M.: From Gauss to Painlevé; A modern theory of special functions. (1991) · Zbl 0743.34014
[6] Vaserstein, L.; Wheland, E.: Products of conjugacy classes of two by two matrices. Linear algebra appl. 230, 165-188 (1995) · Zbl 0840.20040
[7] Vaserstein, L.; Wheland, E.: Rigid relations in GL2F. Linear algebra appl. 281, 25-31 (1998) · Zbl 0944.20028