Stratifying systems and $g$-vectors

TL;DR

This paper investigates the relationship between $g$-vectors and the Cartan matrix in stratifying systems induced by $ au$-rigid modules, providing formulas for the Cartan group and characterizations of diagonal cases.

Contribution

It introduces a method to compute the Cartan group directly from $g$-vectors and characterizes stratifying systems with diagonal Cartan matrices derived from $ au$-rigid modules.

Findings

Cartan group can be computed using $g$-vectors of summands.

Characterization of stratifying systems with diagonal Cartan matrices.

Provides explicit formulas linking $g$-vectors and Cartan matrices.

Abstract

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.