Scalar extensions of quiver representations over $\mathbb{F}_1$
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915188816478208 |
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| author | Kleinau, Markus |
| author_facet | Kleinau, Markus |
| contents | Let $V$ and $W$ be quiver representations over $\mathbb{F}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of $\mathrm{Hom}_{KQ}(V^K,W^K)$ generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04597 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scalar extensions of quiver representations over $\mathbb{F}_1$ Kleinau, Markus Representation Theory Combinatorics Let $V$ and $W$ be quiver representations over $\mathbb{F}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of $\mathrm{Hom}_{KQ}(V^K,W^K)$ generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko. |
| title | Scalar extensions of quiver representations over $\mathbb{F}_1$ |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2403.04597 |