$M$-TF equivalences on the real Grothendieck groups

TL;DR

This paper introduces the $M$-TF equivalence in the context of abelian length categories, showing it forms a rational fan structure related to Newton polytopes, thereby generalizing and completing known fan structures in representation theory.

Contribution

It systematically defines the $M$-TF equivalence, proves its associated set forms a rational generalized fan, and relates this to Newton polytopes, extending the understanding of wall-chamber structures.

Findings

$ ext{Sigma}(M)$ is a finite, complete rational generalized fan.

$ ext{Sigma}(M)$ is the normal fan of the Newton polytope $ ext{N}(M)$.

In module categories, $ ext{Sigma}(M)$ completes a coarsening of the $g$-fan of the algebra.

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 $\theta \in K_0(\mathcal{A})_\mathbb{R}^*$. 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 $\Sigma(M)$ of closures of $M$-TF equivalence classes is a rational generalized fan in $K_0(\mathcal{A})_\mathbb{R}^*$ which is finite and complete. More precisely, we show that $\Sigma(M)$ is the normal generalized fan of the Newton polytope $\mathrm{N}(M)$ in $K_0(\mathcal{A})_\mathbb{R}$. When $\mathcal{A}$ is the category of finitely generated modules over a finite dimensional algebra $A$, $\Sigma(M)$ can be regarded as a completion of a certain coarsening of the $g$-fan of $A$.