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arXiv · 2610.14814

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 $s\to t$, 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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Jindong Yan, Shenglin Zhu. 2026-10-10. A Counterexample To The TF-cone Conjecture. https://arxiv.org/abs/2610.14814

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