The wall-chamber structures of the real Grothendieck groups

For a finite-dimensional algebra A over field K with n simple modules, the real Grothendieck group 0 ( proj ) R : = ? Z ? gives stability conditions of King. We study associated wall-chamber structure by using Koenig–Yang correspondences in silting theory. First, we introduce an equivalence relation on called TF numerical torsion pairs Baumann–Kamnitzer–Tingley. Second, show that open cone spanned g-vectors each 2-term object class, and this one-to-one correspondence between basic objects classes full dimension. Finally, determine case is path acyclic quiver.