Tree representations of the quiver $\widetilde{\mathbb{D}}_{m}$
Volume 167 / 2022
Abstract
We explicitly describe field independent tree representations of the canonically oriented quiver $\widetilde{\mathbb{D}} _{m}$. Recall that matrices of tree representations involve 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 $\widetilde{\mathbb{D}} _{6}$ and a method to obtain tree representations for exceptionals in the canonically oriented general case $\widetilde{\mathbb{D}} _{m}$ from that list. 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 an open question raised/suggested by Ringel also in the $\widetilde{\mathbb{D}} _{m}$ case.
