Let $G_2$ denote the exceptional Lie group $G_2$ as a $\mathbb{Q}$-algebraic group. Suppose that is also given a matrix representation $\rho : G_2\rightarrow SO(7)$. Let $M$ be a matrix with integral coefficients in the image of this representation. What would be an approach to prove that there is no $S$, with integral coefficients, in the image of this representation, such that $M=S^k$. My first attempt was to use jordan decomposition $M = M_s\cdot M_u$ and reduce to the semi-simple case, but that didn't work. Any help is welcome. Thanks!
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4$\begingroup$ When you say "matrix in $G_2$", do you have a specific representation of $G_2$ in mind? I ask because I'm not sure I know how to answer the question already for $SL_n$, much less $G_2$. $\endgroup$– Allen KnutsonFeb 17, 2016 at 3:31
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$\begingroup$ To reinforce Allen's comment, I'd also request a more precise statement. What exactly is meant by $G_2(Z)$, and what are "integral matrices" in $G_2$ other than elements of $G_2(Z)$? $\endgroup$– Jim HumphreysFeb 17, 2016 at 14:06
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$\begingroup$ @AllenKnutson I'm sorry for the confusing question, I've clarified it now. Thanks. $\endgroup$– juniorFeb 17, 2016 at 23:23
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$\begingroup$ @JimHumphreys I'm sorry for the confusing question, I've clarified it now. Thanks. $\endgroup$– juniorFeb 17, 2016 at 23:23
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