Tree representations of the quiver $\widetilde{\mathbb{E}}_{6}$
Volume 164 / 2021
Abstract
We explicitly describe tree representations of the canonically oriented quiver $\widetilde {\mathbb {E}}_{6}$. Recall that tree representations can be exhibited using matrices involving only the elements $0$ and $1$ and the total number of ones is exactly $d-1$ where $d$ is the length of the module. Due to a result of Ringel (1998) the existence of tree representations is guaranteed when the module is exceptional (indecomposable and without self-extensions). In this paper we give a complete and general list corresponding to exceptional modules over the path algebra of the canonically oriented Euclidean quiver $\widetilde {\mathbb {E}}_{6}$. The proof (involving induction and symbolic computation with block matrices) was partially generated by a purposefully developed computer software and is available on arXiv as an appendix to this paper. All the representations given here remain valid over any base field, answering a question raised/suggested by Ringel.
