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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.10055 |
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| _version_ | 1866912072006107136 |
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| author | Gallup, Nathaniel Sawin, Stephen |
| author_facet | Gallup, Nathaniel Sawin, Stephen |
| contents | We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if $Ω$ is any connected quiver, the category of locally finite-dimensional representations of $Ω$ has unique representation type (meaning no two indecomposable representations have the same dimension vector) if and only if the underlying graph of $Ω$ is a generalized ADE Dynkin diagram (i.e. one of $A_n, D_n, E_6, E_7, E_8, A_{\infty}, A_{\infty , \infty}$ or $D_\infty$). This result is companion to earlier work of the authors generalizing Gabriel's theorem to infinite quivers with different conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10055 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers Gallup, Nathaniel Sawin, Stephen Representation Theory 16G20, 16D70, 16G60, 17B22, 06A06 We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if $Ω$ is any connected quiver, the category of locally finite-dimensional representations of $Ω$ has unique representation type (meaning no two indecomposable representations have the same dimension vector) if and only if the underlying graph of $Ω$ is a generalized ADE Dynkin diagram (i.e. one of $A_n, D_n, E_6, E_7, E_8, A_{\infty}, A_{\infty , \infty}$ or $D_\infty$). This result is companion to earlier work of the authors generalizing Gabriel's theorem to infinite quivers with different conditions. |
| title | Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers |
| topic | Representation Theory 16G20, 16D70, 16G60, 17B22, 06A06 |
| url | https://arxiv.org/abs/2410.10055 |