Let $A$ be the path algebra of the quiver $\tilde{D}\_4$. I would like to find its exceptional regular representations with as little computation as possible.


Of course, we can compute the whole Auslander-Reiten quiver and compute all the $\text{Hom}\_A(X,X)$ and $\text{Ext}\_A^1(X,X)$ for all the regular representations, but surely there must be a better way to do it?


A conjecture I have is that an exceptional indecomposable representation must be on the border of the regular component of the AR-quiver. By the border I mean the set of vertices of the quiver such that there is only one incoming and one outgoing edge. If anyone knows a standard term for this set, please tell me in the comments! I think there are only 6 exceptional regular representations of $\tilde{D}\_4$, all of which are of this form. This conjecture seems plausible to me, but I haven't managed to prove this in general.


EDIT: it seems that it's very much untrue that an exceptional module must be on the border of a regular component, but I think I managed to prove this for the components on which the Auslander-Reiten translate $\tau$ satisfies $\tau^2=\text{id}$, and with that it is indeed possible to find the exceptional modules without computing the whole quiver. I will write down an answer here as soon as I find the time.



Finding exceptional regular representations of $\tilde{D}_4$ efficiently
The AR quiver of the regular representations of an affine quiver consists of infinitely many "tubes". A tube of rank $r$ has $r$ modules on what you call the border. Let me number them $B\_1, B\_2, \dots, B\_r$. Then, if I numbered them in the convenient way, there are another $r$ modules $C\_1, \dots C\_r$, and irreducible morphisms from $B\_i$ to $C\_i$ and from $C\_i$ to $B\_{i+1}$. Then there are another $r$ modules $D\_i$ with morphisms from $C\_i$ to $D\_i$ and from $D\_i$ to $C\_{i+1}$. This pattern continues infinitely. The Auslander--Reiten translation sends the module on each level numbered $i$ to the object numbered $i-1$ (cyclically).


The AR quiver of an affine quiver consists of infinitely many rank 1 tubes and up to three tubes of higher rank. The tubes are standard, meaning that you can calculate Homs by looking at compositions of arrows in the component modulo mesh relations. $\operatorname{Ext}^1(V,V)$ is dual to $\operatorname{Hom}(V,\tau V)$, so its dimension can be calculated the same way. One discovers that the exceptional representations in a tube of rank $r$ lie in the bottom $r-1$ layers of the tube.


In $\widetilde D\_n$, there are three exceptional tubes, of rank 2, 2, and $n-2$. This provides $2 + 2 + (n-2)(n-3) = n^2 -5n + 10$ exceptional representations.


A possible reference is Chapter 7 of the book "An introduction to quiver representations" by Derksen and Weyman. Though they say they don't give full details and encourage the reader to consult "Indecomposable graphs and algebras" by Dlab and Ringel, published in the Memoirs of the AMS.


Another excellent reference would be [the notes by Crawley-Boevey](https://www.math.uni-bielefeld.de/%7Ewcrawley/quivlecs.pdf). The answer for $\widetilde D\_4$ is given on page 35.


