$M$-TF equivalences on the real Grothendieck groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910042991624192 |
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| author | Asai, Sota Iyama, Osamu |
| author_facet | Asai, Sota Iyama, Osamu |
| contents | 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 rational generalized fan in $K_0(\mathcal{A})_\mathbb{R}^*$ which is finite and complete. More precisely, we show that $Σ(M)$ 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 $A$, $Σ(M)$ can be regarded as a completion of a certain coarsening of the $g$-fan of $A$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_13232 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $M$-TF equivalences on the real Grothendieck groups Asai, Sota Iyama, Osamu Representation Theory 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 rational generalized fan in $K_0(\mathcal{A})_\mathbb{R}^*$ which is finite and complete. More precisely, we show that $Σ(M)$ 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 $A$, $Σ(M)$ can be regarded as a completion of a certain coarsening of the $g$-fan of $A$. |
| title | $M$-TF equivalences on the real Grothendieck groups |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2404.13232 |