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Tree representations of the quiver $\widetilde{\mathbb{E}}_{6}$

Volume 164 / 2021

Szabolcs Lénárt, Ábel Lőrinczi, István Szöllősi Colloquium Mathematicum 164 (2021), 221-250 MSC: Primary 16G20; Secondary 16G70. DOI: 10.4064/cm7931-1-2020 Published online: 18 September 2020

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.

Authors

  • Szabolcs LénártBitdefender S.R.L.
    Cuza Vodă 1
    400107 Cluj-Napoca, Romania
    e-mail
  • Ábel LőrincziFaculty of Mathematics and Computer Science
    Babeş-Bolyai University
    M. Kogălniceanu 1
    400084 Cluj-Napoca, Romania
    e-mail
  • István SzöllősiFaculty of Mathematics and Computer Science
    Babeş-Bolyai University
    M. Kogălniceanu 1
    400084 Cluj-Napoca, Romania
    and
    Faculty of Informatics
    Eötvös Loránd University
    Pázmány P. 1/C
    H-1117 Budapest, Hungary
    e-mail

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