# The explicit indecomposable representations of (any) Euclidean quiver of type E

It is known that for any quiver $Q$ that is an orientation of $\tilde{\mathbb{E}}_8$, the hereditary path algebra $KQ$ ($K$ being an algebraically closed field) is tame (but not finite). That is, in each dimension $d$, all but a finite number of indecomposable $KQ$-modules occur in a finite number of 1-parameter families. (These 1-parameter families arise from $K[t]$-$KQ$-bimodules that are finitely generated and free left $K[t]$-modules.)

I am interested in the explicit quiver representations of these 1-parameter families of $KQ$-modules.

An example for the orientation $$\begin{matrix} & & \uparrow & \\ \leftarrow & \leftarrow & & \rightarrow & \rightarrow & \rightarrow & \rightarrow & \rightarrow \end{matrix}$$ is provided at the top of page 187 in the following reference:

Ibrahim Assem, Daniel Simson, and Andrzej Skowronski, MR 2197389 Elements of the representation theory of associative algebras. Vol. 1, ISBN: 978-0-521-58423-4; 978-0-521-58631-3; 0-521-58631-3.

However, I do not see anything on the other orientations of $\tilde{\mathbb{E}}_8$. I would like to know a specific example of a 1-parameter family of indecomposable $KQ$-modules for every other orientation of $\tilde{\mathbb{E}}_8$. In my experience, whilst the dimension vectors of 'similar' families of indecomposable modules may be the same across different orientations, the linear maps in the $K$-representations usually need to be completely different to maintain indecomposability.

I would like a reference (if one exists) which lists such examples (and I really do need an explicit presentation). Alternatively, if there exists a technique to construct these families of modules for any orientation, a reference for this would also be appropriate.

• There are 256 different orientations, so I don't think it's surprising that you can't find descriptions written out explicitly for for every orientation. Do you know about "reflection functors" (if not, Google)? They'll translate a 1-parameter family for any orientation to one for any other orientation. – Jeremy Rickard May 12 '17 at 19:45

Crawley-Boevey's notes (http://www1.maths.leeds.ac.uk/~pmtwc/quivlecs.pdf) show how to get a family of indecomposable modules indexed by $\mathbb{P}^1$ by a general method. Consider the projective $P$ covering the simple for the extending vertex (this is itself simple in the example above, but in other orientations it might not be), and let $L$ be the unique indecomposable module whose dimension vector is that of $P$ plus $\delta$. Some simple calculations show that $\mathrm{Hom}(P,L)$ is a 2-dimensional vector space and every non-zero element of this space gives an injective map, with cokernel given by an indecomposable (simple in all but finitely many cases) representation with dimension vector $\delta$. Throwing out the finitely many non-simple points, this is the unique 1-dimensional family of simple representations with this dimension vector.