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Suppose there is a group-homomorphism $f: G \rightarrow \operatorname{GL}(n,K)$ with $a\mapsto A, b\mapsto B$ for a group $G$ which is generated by two elements $a,b$ and matrices $A,B \in \operatorname{GL}(n,K)$. How can you calculate here the $f$-invariant subspaces of $K^n$?

My problem here is, that there are two matrices $A$ and $B$. If $f$ would be defined just by one matrix $A$ than I can get the $f$-invariant subspaces by calculating the generalized eigenspaces (for n=2 just the normal eigenspaces) of $A$. What do I have to do when there is a second matrix $B$ and why?

Could someone help me please.

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    You want $f$ to be a group homomorphism, right?2012-08-21
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    yes, I will add it above2012-08-21
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    Guess you should add a representation theory tag. This might even be the answer you are looking for.2012-08-21
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    Can you explain "I can get the $f$-invariant subspaces by calculating the generalized eigenspaces"? I think you mean $A$-invariant, or $f(a)$-invariant. However the problem is quite subtle here already. For instance take $A$ a unipotent matrix, there is only the whole space that is generalized eigenspace for $1$, but the set of invariant subspaces is not so easy to describe (and depends on further details of $A$). For the two-matrix problem, you are looking for subspaces that are both $A$-invariant and $B$-invariant, right?2012-08-22

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