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Consider the quiver $1 \overset{\alpha}{\underset{\alpha'}{\leftleftarrows}} 2$. I am trying to find the form of the Auslander–Reiten quiver. So I got :

$P(1)= \begin{matrix} 1 \end{matrix}$, $P(2) =\begin{matrix} 2 &\\ 1 & 1 \end{matrix}$, $I(1) =\begin{matrix} 2 & 2\\ 1 & \end{matrix}$, $I(2)= \begin{matrix} 2 \end{matrix}$, $S(1)=1$, $S(2)=2$. I am just curious about how the Auslander–Reiten quiver looks like.

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    $\begingroup$ You should tell us a bit what your background is. Have you computed any AR. quiver at all? $\endgroup$ Commented Sep 13, 2015 at 3:13
  • $\begingroup$ (That algebra is of infinite representation type, so it is probably not the very first example to try...) $\endgroup$ Commented Sep 13, 2015 at 3:14
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    $\begingroup$ Well, you can pretty much do exactly the same thing with this quiver as with those! Start with the simple projective and keep knitting the quiver. $\endgroup$ Commented Sep 13, 2015 at 5:06
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    $\begingroup$ They cannot be "the same" because they are reprentations of different quivers. over the Kronecker quiver (the one with two parallel arrows) one of the projectives is simple, equal to S(1), while the other has dimension $3$, and its composition factors are S(1), S(1) and S(2). $\endgroup$ Commented Sep 13, 2015 at 5:24
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    $\begingroup$ I suggest you review all this before embarking in computing AR-quivers, as it is pretty fundamental! $\endgroup$ Commented Sep 13, 2015 at 5:24

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The result is too long to post here so I just give a reference. There are two approaches to find the Auslander-Reiten quiver (over an algebraically closed field): One more or less just linear algebra based (and thus elementary) approach is described in the book of Barot: Introduction to the Representation theory of algebras Chapter 6.2. As a special result about the representation-theory of tame hereditary algebras, the Auslander-Reiten quiver is computed very fast and nice in the Book of Simson and Skowronski: Elements of the Representation Theory of Associative Algebras: Volume 2. If you dont want to assume that the field is algebraically closed, then Chapter 4.5 in the first volume of the books by Benson helps you and gives a relation to the indecomposable representations of the klein four group over a field of characteristic 2.

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