On the Orthogonal Projections
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913089846247424 |
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| author | Fei, Jiarui |
| author_facet | Fei, Jiarui |
| contents | For any rigid presentation $e$, we construct an orthogonal projection functor to ${\rm rep}(e^\perp)$ left adjoint to the natural embedding. We establish a bijection between presentations in ${\rm rep}(e^\perp)$ and presentations compatible with $e$. For quivers with potentials, we show that ${\rm rep}(e^\perp)$ forms a module category of another quiver with potential. We derive mutation formulas for the $δ$-vectors of positive and negative complements and the dimension vectors of simple modules in ${\rm rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Orthogonal Projections Fei, Jiarui Representation Theory Rings and Algebras Primary 16G10, Secondary 13F60 For any rigid presentation $e$, we construct an orthogonal projection functor to ${\rm rep}(e^\perp)$ left adjoint to the natural embedding. We establish a bijection between presentations in ${\rm rep}(e^\perp)$ and presentations compatible with $e$. For quivers with potentials, we show that ${\rm rep}(e^\perp)$ forms a module category of another quiver with potential. We derive mutation formulas for the $δ$-vectors of positive and negative complements and the dimension vectors of simple modules in ${\rm rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors. |
| title | On the Orthogonal Projections |
| topic | Representation Theory Rings and Algebras Primary 16G10, Secondary 13F60 |
| url | https://arxiv.org/abs/2510.01615 |