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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.11946 |
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| _version_ | 1866917593214877696 |
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| author | Lin, Zengqiang Su, Xiuping |
| author_facet | Lin, Zengqiang Su, Xiuping |
| contents | Associated to a symmetrisable Cartan matrix $C$, Geiss-Lerclerc-Schröer constructed and studied a class of Iwanaga-Gorenstein algebras $H$. They proved a generalised version of Gabriel's Theorem, that is, the rank vectors of $τ$-locally free $H$-modules are the positive roots of type $C$ when $C$ is of finite type, and conjectured that this is true for any $C$. In this paper, we look into this conjecture when $C$ is of affine type. We construct explicitly stable tubes, some of which have rigid mouth modules, while others not. We deduce that any positive root of type $C$ is the rank vector of some $τ$-locally free $H$-module. However, the converse is not true in general. Our construction shows that there are $τ$-locally free $H$-modules whose rank vectors are not roots, when $C$ is of type $\widetilde{\mathbb{B}}_n$, $\widetilde{\mathbb{CD}}_n$, $\widetilde{\mathbb{F}}_{41}$ and $\widetilde{\mathbb{G}}_{21}$, and so the conjecture fails in these four types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11946 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Affine root systems, stable tubes and a conjecture by Geiss-Leclerc-Schröer Lin, Zengqiang Su, Xiuping Representation Theory 16G10, 16G20, 16G70 Associated to a symmetrisable Cartan matrix $C$, Geiss-Lerclerc-Schröer constructed and studied a class of Iwanaga-Gorenstein algebras $H$. They proved a generalised version of Gabriel's Theorem, that is, the rank vectors of $τ$-locally free $H$-modules are the positive roots of type $C$ when $C$ is of finite type, and conjectured that this is true for any $C$. In this paper, we look into this conjecture when $C$ is of affine type. We construct explicitly stable tubes, some of which have rigid mouth modules, while others not. We deduce that any positive root of type $C$ is the rank vector of some $τ$-locally free $H$-module. However, the converse is not true in general. Our construction shows that there are $τ$-locally free $H$-modules whose rank vectors are not roots, when $C$ is of type $\widetilde{\mathbb{B}}_n$, $\widetilde{\mathbb{CD}}_n$, $\widetilde{\mathbb{F}}_{41}$ and $\widetilde{\mathbb{G}}_{21}$, and so the conjecture fails in these four types. |
| title | Affine root systems, stable tubes and a conjecture by Geiss-Leclerc-Schröer |
| topic | Representation Theory 16G10, 16G20, 16G70 |
| url | https://arxiv.org/abs/2402.11946 |