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Consider the following two quivers:

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I have two questions.

  1. Are there only finitely many nonisomorphic representations of $\mathbb{C}Q$ with dimension vector $(2, 1, 1, 1)$?
  2. Are there only finitely many nonisomorphic representations of $\mathbb{C}Q'$ with dimension vector $(2, 1, 1, 1, 1)$?

In both cases, the $2$-dimensional space is placed at the central vertex.

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  • $\begingroup$ Q is dynkin of finite type and so has only finely many. $\endgroup$ Oct 24, 2015 at 22:36
  • $\begingroup$ Q' is tame so it should be possible to answer your question using the classification of indecomposables $\endgroup$ Oct 24, 2015 at 22:39
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    $\begingroup$ en.wikipedia.org/wiki/Cross-ratio $\endgroup$ Oct 25, 2015 at 3:00
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    $\begingroup$ If I understood @AllenKnutson correctly you can think of your representation as choosing 4 lines in a 2-dimensional space and the cross ratio is an isomorphism invariant so infinitely many isomorphism classes. $\endgroup$ Oct 25, 2015 at 15:56

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To expand on Allen's answer: the isomorphism type of such a representation is determined just by the images of the incoming maps, which must each be a line if the rep is indecomposable, up to the action of linear transformations on the 2 dimensional space.

  • the moduli space in indecomposable representations in the former case (ignoring automorphisms) is a single point: any 3 linearly independent 1-d subspaces in a 2 dimensional space (i.e. any 3 distinct points in $\mathbb{P}^1$) can be sent to any other by a linear transformation. This linear transformation is actually unique up to scalar multiplication.

  • the moduli space in the latter case is $\mathbb{P}^1$ minus 3 points, and the isomorphism is cross ratio: you use the isomorphism above to send 3 of the lines (say, from the NE, NW and SE points) to $\{0,1,\infty\}$, and you still have an arbitrary point (from the SW) left over.

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