Shard theory for $g$-fans
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912046706065408 |
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| author | Mizuno, Yuya |
| author_facet | Mizuno, Yuya |
| contents | For a finite dimensional algebra $A$, the notion of $g$-fan $Σ(A)$ is defined from two-term silting complexes of $A$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$. In this paper, we discuss the theory of shards to $Σ(A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of the poset of torsion classes of $\mathrm{mod} A$ and the set of shards of $Σ(A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $\mathrm{mod} A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $\mathrm{mod} A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_10745 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Shard theory for $g$-fans Mizuno, Yuya Representation Theory Combinatorics For a finite dimensional algebra $A$, the notion of $g$-fan $Σ(A)$ is defined from two-term silting complexes of $A$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$. In this paper, we discuss the theory of shards to $Σ(A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of the poset of torsion classes of $\mathrm{mod} A$ and the set of shards of $Σ(A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $\mathrm{mod} A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $\mathrm{mod} A$. |
| title | Shard theory for $g$-fans |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2212.10745 |