Proof of the tree module property for exceptional representations of tame quivers

TL;DR

This paper provides computer-assisted proofs for the tree module property of exceptional representations over specific Euclidean quivers, using symbolic computation and induction, with detailed proofs and an overview of the computational methods.

Contribution

It introduces a computer-aided proof technique for the tree module property in Euclidean quivers, including detailed step-by-step computational proofs and a theoretical overview.

Findings

Complete list of tree representations for exceptional modules over $ ilde{E}_6$ and $ ilde{D}_6$ quivers.

Computer-generated proofs involving symbolic computation and induction.

Methodology combining theoretical and computational approaches for quiver representations.

Abstract

This document serves as an arXiv entry point for the appendix to the paper [13] (the ancillary file e6_proof.pdf -- ``Proof of the tree module property for exceptional representations of the quiver $\widetilde{\mathbb{E}}_6$'') and the appendix to the paper [12] (the ancillary file d6_proof.pdf -- ``Proof of the tree module property for exceptional representations of the quiver $\widetilde{\mathbb{D}}_6$''). The ancillary files contain the computer generated part of the proofs of the main results in [13] respectively [12], giving a complete and general list of tree representations corresponding to exceptional modules over the path algebra of the canonically oriented Euclidean quiver $\widetilde{\mathbb{E}}_6$, respectively $\widetilde{\mathbb{D}}_6$. The proofs (involving induction and symbolic computation with block matrices) were partially generated by a purposefully developed computer software, outputting in a detailed step-by-step fashion as if written ``by hand''. We also give here a short theoretical introduction and an overview of the computational method used to prove the formulas given in the papers [13] and [12].