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Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying
$$I+A+\cdots+A^n=0.$$ It is easy to dope out some obvious solutions.Say $(-I,1)$.But how to find all the solutions to this equation?

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    $\begingroup$ Your equation is equivalent to $A^{n+1}=I$ and $A\neq I$. $\endgroup$
    – Benoît Kloeckner
    Commented Jan 11, 2014 at 13:40
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    $\begingroup$ This website is for math research, and I'm not certain your question qualifies. A good start can be made, anyway, by multiplying your equation by $I-A$. Finding matrices with $A^m=I$ has a literature. $\endgroup$
    – Gerry Myerson
    Commented Jan 11, 2014 at 13:40
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    $\begingroup$ Cross-posted to math.stackexchange $\endgroup$
    – Marc van Leeuwen
    Commented Jan 11, 2014 at 14:16
  • $\begingroup$ I'm not sure that it's that simple as @lennon310 suggests and that the question does not qualify. As far as I know, integral representations of finite cyclic groups are still an open problem. The only hope is the small dimension. I would suggest (but not sure) that there are finitely many conjugacy classes. It is relatively easy to describe them in $GL_3(\mathbb{Q}[i])$, but the passage to $\mathbb{Z}[i]$ may be a bigger problem. $\endgroup$
    – Alex Degtyarev
    Commented Jan 11, 2014 at 16:03
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    $\begingroup$ @Benoît Kloeckner: not quite true: instead of $A^n\ne I$ one should require that $A$ has no invariant vectors. $\endgroup$
    – Alex Degtyarev
    Commented Jan 11, 2014 at 16:05

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