Functor-induced isomorphisms and $G$-matrices

In this paper, we explore how functor-induced isomorphisms are encoded by $G$-matrices. We first show that the Grothendieck group isomorphism induced by a tilting module can be realized via the $G$-matrix of this tilting module. Building on this, we compare $g$-vectors for a tilted algebra and its associated hereditary algebra, and provide $G$-matrix interpretations of the Coxeter transformation, the Nakayama functor, and the Auslander-Reiten translation for suitable algebras. Furthermore, we demonstrate that every element of any symmetric group and Weyl group can be expressed as the transpose of the $G$-matrix of some tilting module or support $\tau$-tilting module. Finally, we show that the Grothendieck group isomorphism induced by a $2$-term silting complex can also be realized via the $G$-matrix of this $2$-term silting complex.