### Speaker

### Description

This talk is based on joint work with Osamu Iyama. The representation theory of a finite dimensional algebra $A$ deals with the category $\mathsf{mod} A$ of finitely generated $A$-modules. One of the main topics is torsion pairs in $\mathsf{mod} A$. Functorially finite torsion pairs have been well-studied, but they are too few among all torsion pairs. Thus, we are now studying a wider class called semistable torsion pairs introduced by Baumann-Kamnitzer-Tingley associated to elements of the real Grothendieck group $K_0(\mathsf{proj}A)_\mathbb{R}$ of the category of finitely generated projective $A$-modules, which is identified with the Euclidean space whose canonical basis is given by the isoclasses of indecomposable projective $A$-modules. By using semistable torsion pairs, I (Asai) introduced an equivalence relation called TF equivalence on $K_0(\mathsf{proj} A)_\mathbb{R}$. A typical example of TF equivalence classes is the silting cone $C^\circ(U)$ generated by the g-vectors of indecomposable direct summands of each 2-term presilting complex $U$. On the other hand, there can be other TF equivalence classes. To study them, we have found that canonical decompositions introduced by

Derksen-Fei is useful. I would like to explain some important properties of these notions.