Stratifying systems and $g$-vectors

In this paper we study the Cartan matrix associated to the Ext-projective stratifying system induced by a basic and $\tau$-rigid object $M$ in mod$(A)$ by means of the $g$-vectors of the indecomposable direct summands of $M$. In particular we show that the Cartan group of a stratifying system associated to a $\tau$-rigid module can be calculated directly using these vectors. Moreover we characterise the stratifying systems coming from $\tau$-rigid modules that have a diagonal Cartan matrix.