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PreprintarXiv licencearXiv

A Counterexample To The TF-cone Conjecture

  • Jindong Yan
  • Shenglin Zhu
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Abstract

The TF-cone conjecture of Asai and Iyama predicts that the TF equivalence class of an integral g-vector is the open cone generated by the canonical summands of all its positive multiples. We give a counterexample to this conjecture. For the tensor product of the three-arrow Kronecker algebra with the path algebra of s→ t, a vector w has a two-dimensional TF equivalence class, whereas every positive integral multiple mw is generically indecomposable. The cone generated by these canonical summands is therefore a single ray. The same example disproves the relative-interior version proposed by Haerizadeh and Yassemi.

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Cite

BibTeXRIS
BibTeX
@misc{yan2026counterexample,
  title = {{A Counterexample To The TF-cone Conjecture}},
  author = {Jindong Yan and Shenglin Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14814},
  eprint = {2610.14814},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14814},
}
RIS
TY  - UNPB
TI  - A Counterexample To The TF-cone Conjecture
AU  - Jindong Yan
AU  - Shenglin Zhu
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.14814
UR  - https://arxiv.org/abs/2610.14814
AB  - The TF-cone conjecture of Asai and Iyama predicts that the TF equivalence class of an integral g-vector is the open cone generated by the canonical summands of all its positive multiples. We give a counterexample to this conjecture. For the tensor product of the three-arrow Kronecker algebra with the path algebra of s→ t, a vector w has a two-dimensional TF equivalence class, whereas every positive integral multiple mw is generically indecomposable. The cone generated by these canonical summands is therefore a single ray. The same example disproves the relative-interior version proposed by Haerizadeh and Yassemi.
ER  -