Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras

Abstract

In the representation theory of finite-dimensional algebras $A$ over a field, the classification of 2-term (pre)silting complexes is an important problem. One of the useful tool is the g-vector cones associated to the 2-term presilting complexes in the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_{\mathbb{Z}} {\mathbb{R}}$. The aim of this paper is to study the complement $\operatorname{\mathsf{NR}}$ of the union $\operatorname{\mathsf{Cone}}$ of all g-vector cones, which we call the non-rigid region. By the work of Iyama and us, $\operatorname{\mathsf{NR}}$ is determined by 2-term presilting complexes and a certain closed subset $R_0 \subset K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}$, which is called the purely non-rigid region. In this paper, we give an explicit description of $R_0$ for complete special biserial algebras in terms of a finite set of maximal nonzero paths in the Gabriel quiver of $A$. We also prove that $\operatorname{\mathsf{NR}}$ has some kind of fractal property and that $\operatorname{\mathsf{NR}}$ is contained in a union of countably many hyperplanes of codimension one. Thus, any complete special biserial algebra is g-tame, that is, $\operatorname{\mathsf{Cone}}$ is dense in $K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}$.