An algorithm for constructing the AR-quiver of a path algebra corresponding to a change in the orientation. - MathOverflow

An algorithm for constructing the AR-quiver of a path algebra corresponding to a change in the orientation.

2012-01-28T22:27:41Z

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.

Answer by Steve for An algorithm for constructing the AR-quiver of a path algebra corresponding to a change in the orientation.

2012-01-29T02:13:29Z

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.