The wall-chamber structures of the real Grothendieck groups

2019/05/06 by Sota Asai, Asai, Sota · 6 citations
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1905.02180

Abstract

For a finite-dimensional algebra A over a field K with n simple modules, the real Grothendieck group K₀(proj A)_ℝ:=K₀(proj A) ⊗_ℤ ℝ ≅ ℝⁿ gives stability conditions of King. We study the associated wall-chamber structure of K₀(proj A)_ℝ by using the Koenig--Yang correspondences in silting theory. First, we introduce an equivalence relation on K₀(proj A)_ℝ called TF equivalence by using numerical torsion pairs of Baumann--Kamnitzer--Tingley. Second, we show that the open cone in K₀(proj A)_ℝ spanned by the g-vectors of each 2-term silting object gives a TF equivalence class, and this gives a one-to-one correspondence between the basic 2-term silting objects and the TF equivalence classes of full dimension. Finally, we determine the wall-chamber structure of K₀(proj A)_ℝ in the case that A is a path algebra of an acyclic quiver.

Cited by

• Finite type and completeness of g-fans
• Stable Brauer-Thrall II' conjecture for finite-dimensional Jacobian algebras
• Fans and polytopes in tilting theory III: Classification of convex g-fans of rank 3
• Bricks and τ-tilting theory under base field extensions
• Two-term tilting complexes for preprojective algebras of non-Dynkin type
• Finite-dimensional Jacobian algebras: Finiteness and tameness

Related

• M-TF equivalences on the real Grothendieck groups
• Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras
• A silting theorem
• Silting complexes and Gorenstein projective modules
• Caracteres de rigidite du groupe de Grothendieck-Teichmuller