$M$-TF equivalences on the real Grothendieck groups
Sota Asai, Osamu Iyama
TL;DR
The paper addresses stability-wall structures in abelian length categories by introducing $M$-TF equivalence, a targeted coarsening of TF equivalence parameterized by an object $M$. It constructs, for each $M$, a finite complete fan $\\boldsymbol{\\Sigma}(M)$ and proves it coincides with the normal fan of the Newton polytope $\\mathrm{N}(M)$, yielding a precise convex-geometric description of $M$-TF classes. The work provides explicit examples showing nuanced behavior for non-brick $M$ and connects torsion-pair data to polyhedral faces, establishing a robust framework to analyze walls, stability, and decompositions in representation-theoretic settings. This framework interfaces with cluster theory and tau-tilting theory, offering concrete combinatorial tools for understanding stability conditions and their modulated equivalence classes.
Abstract
For an abelian length category $\mathcal{A}$ with only finitely many isoclasses of simple objects, we have the wall-chamber structure and the TF equivalence on the dual real Grothendieck group $K_0(\mathcal{A})_\mathbb{R}^*=\operatorname{Hom}_\mathbb{R}(K_0(\mathcal{A})_\mathbb{R},\mathbb{R})$, which are defined by semistable subcategories and semistable torsion pairs in $\mathcal{A}$ associated to elements $θ\in K_0(\mathcal{A})_\mathbb{R}^*$. In this paper, we introduce the $M$-TF equivalence for each object $M \in \mathcal{A}$ as a systematic way to coarsen the TF equivalence. We show that the set $Σ(M)$ of closures of $M$-TF equivalence classes is a finite complete fan in $K_0(\mathcal{A})_\mathbb{R}^*$, and that $Σ(M)$ is the normal fan of the Newton polytope $\mathrm{N}(M)$ in $K_0(\mathcal{A})_\mathbb{R}$.