# TF equivalence classes constructed from canonical decompositions

This talk is based on joint work with Osamu Iyama. Let $$A$$ be a finite dimensional algebra over an algebraically closed field. Brüstle-Smith-Treffinger introduced a wall-chamber structure on the real Grothendieck group $$K_0(\operatorname{proj} A)_R$$ via stability conditions of King. It is strongly related to TF equivalence, which is an equivalence relation on $$K_0(\operatorname{proj} A)_R$$ defined by numerical torsion pairs of Baumann-Kamnitzer-Tingley. Thanks to results by Yurikusa and Brüstle-Smith-Treffinger, I showed that the $$g$$-vector cone $$C^+(U)$$ associated to each 2-term presilting complex $U$ in $$K^b(\operatorname{proj} A)$$ is a TF equivalence class in my previous study, but we cannot obtain all TF equivalence classes in this way unless $$A$$ is $$\tau$$-tilting finite. In this joint work with Iyama, we obtained a generalization of this construction of TF equivalence classes by using canonical decompositions of elements in $$K_0(\operatorname{proj} A)$$ introduced by Derksen-Fei in the case that $$A$$ satisfies the condition called $$E$$-tameness. I will talk about this result.