The cones of g-vectors

TL;DR

This paper investigates the geometric structure of g-vectors in finite-dimensional algebras, revealing their rational, simplicial nature and conditions for TF-equivalence, thus advancing understanding of wall and chamber structures.

Contribution

It demonstrates that cones of g-vectors are rational and simplicial, and establishes conditions linking open cones and TF-equivalence classes, generalizing prior results.

Findings

G-vectors cones are rational and simplicial.

Open cone and TF-equivalence class interior coincide under certain conditions.

G-vectors satisfy the ray condition far from the origin.

Abstract

This paper studies the wall and chamber structure of algebras via generic decompositions of g-vectors. Specifically, we examine points outside the chambers of the wall and chamber structure of ($\tau$-tilting infinite) finite-dimensional algebras. We demonstrate that the cones of g-vectors are both rational and simplicial. Moreover, we show that the open cone of a given g-vector and the interior of its TF-equivalence class coincide if and only if they are of the same dimension. Furthermore, we establish that g-vectors satisfy the ray condition when sufficiently far from the origin. These results allow us to generalize several findings by Asai and Iyama regarding TF-equivalence classes of g-vectors.