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The cones of g-vectors

Mohamad Haerizadeh, Siamak Yassemi

TL;DR

This work analyzes the wall and chamber structure of finite-dimensional algebras through generic decompositions of $g$-vectors, paying special attention to points outside chambers in the $ au$-tilting infinite setting. It establishes that the cones of $g$-vectors are rational and simplicial, and proves that the open cone $ ext{Cone}^{\

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 ($τ$-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.

The cones of g-vectors

TL;DR

This work analyzes the wall and chamber structure of finite-dimensional algebras through generic decompositions of -vectors, paying special attention to points outside chambers in the -tilting infinite setting. It establishes that the cones of -vectors are rational and simplicial, and proves that the open cone $ ext{Cone}^{\

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 (-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.
Paper Structure (9 sections, 42 theorems, 76 equations, 2 figures)

This paper contains 9 sections, 42 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Preliminaries and background
  4. $\operatorname{TF}$-equivalence classes
  5. Real vectors
  6. g-vectors
  7. Main results
  8. Cones and $\operatorname{TF}$-equivalence classes of g-vectors
  9. Semistable torsion pairs

Key Result

Theorem 1.2

Let $g$ be a tame g-vector. Then

Figures (2)

  • Figure 1: A simulation for the proof of Lemma \ref{['075288509646']}
  • Figure 2: Our idea to prove Theorem \ref{['076273627938']}

Theorems & Definitions (95)

  • Conjecture 1.1
  • Theorem 1.2: \ref{['376103330122']}
  • Theorem 1.3: \ref{['076273627938']} and \ref{['355706191811']}
  • Theorem 1.4: \ref{['962650431975']} and \ref{['401237755092']}
  • Lemma 1.5: \ref{['988807280630']}
  • Proposition 1.6: \ref{['206559753624']}
  • Conjecture 1.7
  • Theorem 1.8: \ref{['948850386317']}
  • Corollary 1.9: \ref{['519973069445']}
  • Definition 3.1
  • ...and 85 more