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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2504.18996 |
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| _version_ | 1866912532752498688 |
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| author | Sengupta, Annoy Kuber, Amit |
| author_facet | Sengupta, Annoy Kuber, Amit |
| contents | For a zero-relation algebra over a field $\mathcal K$, Crawley-Boevey introduced the concept of a tree module and provided a combinatorial description of a basis for the space of homomorphisms between two tree modules--the basis elements are called graph maps. The indecomposability of tree modules is essentially due to Gabriel. We relax a condition in the definition of a tree module to define generalised tree modules and when $\mathrm{char}(\mathcal K)\neq2$, under a certain condition, provide a combinatorial description of a finite generating set for the space of homomorphisms between two such modules--we call the generators generalised graph maps. As an application, we provide a sufficient condition for the (in)decomposability of certain generalised tree modules. We also show that all indecomposable modules over a Dynkin quiver of type $\mathbf D$ are isomorphic to generalised tree modules--this result also follows from a theorem of Ringel which states that all exceptional modules over the path algebra $\mathcal KQ$ of a finite quiver $Q$ are generalised tree modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18996 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalised tree modules: Hom-sets and indecomposability Sengupta, Annoy Kuber, Amit Representation Theory 16G20 For a zero-relation algebra over a field $\mathcal K$, Crawley-Boevey introduced the concept of a tree module and provided a combinatorial description of a basis for the space of homomorphisms between two tree modules--the basis elements are called graph maps. The indecomposability of tree modules is essentially due to Gabriel. We relax a condition in the definition of a tree module to define generalised tree modules and when $\mathrm{char}(\mathcal K)\neq2$, under a certain condition, provide a combinatorial description of a finite generating set for the space of homomorphisms between two such modules--we call the generators generalised graph maps. As an application, we provide a sufficient condition for the (in)decomposability of certain generalised tree modules. We also show that all indecomposable modules over a Dynkin quiver of type $\mathbf D$ are isomorphic to generalised tree modules--this result also follows from a theorem of Ringel which states that all exceptional modules over the path algebra $\mathcal KQ$ of a finite quiver $Q$ are generalised tree modules. |
| title | Generalised tree modules: Hom-sets and indecomposability |
| topic | Representation Theory 16G20 |
| url | https://arxiv.org/abs/2504.18996 |