# 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.