The wall-chamber structures of the real Grothendieck groups

By Sotaro Asai. Advances in Mathematics, 2019

63 citations recorded by the source corpus.

For a finite-dimensional algebra $A$ over a field $K$ with $n$ simple modules, the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_\mathbb{Z} \mathbb{R} \cong \mathbb{R}^n$ gives stability conditions of King. We study the associated wall-chamber structure of $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ by using the Koenig--Yang correspondences in silting theory. First, we introduce an equivalence relation on $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ called TF equivalence by using numerical torsion pairs of Baumann--Kamnitzer--Tingley. Second, we show that the open cone in $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ 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 $K_0(\operatorname{\mathsf{proj}} A)_\mathbb{R}$ in the case that $A$ is a path algebra of an acyclic quiver.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d658f385c56b7664a905903ca064e485f2342748

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In Algebra and Number Theory

  1. C. Pfeifer
  2. Sotaro Asai

In Geometry and Topology

  1. C. Pfeifer
  2. Sotaro Asai

In Mathematical Physics

  1. C. Pfeifer
  2. Sotaro Asai

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