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Considering the path algebra of the quiver $\mathbb{A}_n$, it is well known its Auslander-Reiten quiver with the canonical orientation of $\mathbb{A}_n$, that is, with all the arrows from, say, left to right. I can find as examples in several texts the AR-quivers of $\mathbb{A}_n$ with other orientations.

QUESTION: Is there an algorithm to construct the AR-quiver of $\textit{any}$ orientation of $\mathbb{A}_n$ ?

Clearly it will suffice to describe the effect in the AR-quiver if it is changed the orientation of one arrow. Observing the examples I think there is some pattern, but I can't figure out the algorithm.

Of course this algorithm could be applied to quivers of other type than $\mathbb{A}_n$, but I think it is better to understand this in the simplest case.

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    $\begingroup$ The so called knitting procedure (see the book by Gabriel and Roiter for the problem of who to attribute it to...) will construct the AR-quiver of those algebras in a breeze. This is explained in the book by Gabriel and Roiter, in Assem+Skowroński+Simpson, in ARS iirc, and surely elsewhere. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 29, 2012 at 3:29

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The algorithm for constructing the AR-quiver of any orientation of $A_n$ is the same as the algorithm for constructing the AR-quiver of the orientation you describe. Start out by figuring out all the irreducible maps between the indecomposable projectives. Then write out the cokernels, and then the cokernels of the new maps, and so on until you get to all of the indecomposable injectives.

For example, consider the $A_3$ of the form $1\xleftarrow{} 2\xrightarrow{} 3$. You have two simple projectives $P_1$ and $P_3$ and the non-simple $P_2=(k\xleftarrow{}k\xrightarrow{} k)$. Both $P_1$ and $P_3$ map into $P_2$ with respective cokernels $I_3=(0\xleftarrow{} k\xrightarrow{} k)$ and $I_1=(k\xleftarrow{} k\xrightarrow{} 0)$. The cokernel of $P_2\to I_1\oplus I_3$ is the simple injective $I_2$. So the AR-quiver is like a fish swimming to the right.

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  • $\begingroup$ Thank you Steve, this is certainly an algorithm to construct the AR-quiver directly from the oriented $A_n$-quiver. I was wondering if there is anyone that in each step constructs the AR-quiver corresponding to one-arrow change from the AR-quiver that we had before that change. $\endgroup$
    – Rogelio Fernández-Alonso
    Commented Jan 29, 2012 at 4:06
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    $\begingroup$ This isn't exactly what you are looking for, but I believe you can algorithmically go between the AR-quivers of orientations related by source/sink mutations (also called APR mutations). This corresponds to flipping the direction of all arrows incident to a source or a sink. $\endgroup$
    – Steve
    Commented Jan 29, 2012 at 4:59
  • $\begingroup$ As Steve said: you can pick a sink, which corresponds to a simple projective module. Flipping the arrows at it will remove that node from the AR quiver and add a node at the righthand end of the AR quiver for the new simple injective corresponding to the new source. (And the opposite applies if you start from a source.) To flip a single arrow is harder, but there is always a sequence of sink/source mutations which would do it, so that gives you one approach. $\endgroup$
    – Hugh Thomas
    Commented Feb 2, 2012 at 2:47

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