A geometric realization for maximal almost pre-rigid representations over type $\mathbb{D}$ quivers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912243009978368 |
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| author | Chen, Jianmin Zheng, Yiting |
| author_facet | Chen, Jianmin Zheng, Yiting |
| contents | By using the equivariant theory of group actions, we give a geometric model for the category of finite dimensional representations over a type $\mathbb{D}$ quiver $Q_{D}$ with $n$ vertices and directional symmetry. Furthermore, we introduce the notion of maximal almost pre-rigid representations over $Q_{D}$, which form a family of objects counted by the generalized Catalan number. We present a geometric realization for maximal almost pre-rigid representations and prove that the endomorphism algebras of maximal almost pre-rigid representations are tilted algebras of type $Q_{\overline{D}}$, where $Q_{\overline{D}}$ is a quiver obtained by adding $n-2$ new vertices and $n-2$ arrows to the quiver $Q_{D}$. Additionally, we define a partial order on the set of maximal almost pre-rigid representations, which therefore presents a representation-theoretic interpretation of the type-$\mathbb{D}$ Cambrian lattice determined by $Q_{D}$. Meanwhile, we obtain a representation-theoretic interpretation of the type-$\mathbb{B}$ Cambrian lattices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_03395 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometric realization for maximal almost pre-rigid representations over type $\mathbb{D}$ quivers Chen, Jianmin Zheng, Yiting Representation Theory By using the equivariant theory of group actions, we give a geometric model for the category of finite dimensional representations over a type $\mathbb{D}$ quiver $Q_{D}$ with $n$ vertices and directional symmetry. Furthermore, we introduce the notion of maximal almost pre-rigid representations over $Q_{D}$, which form a family of objects counted by the generalized Catalan number. We present a geometric realization for maximal almost pre-rigid representations and prove that the endomorphism algebras of maximal almost pre-rigid representations are tilted algebras of type $Q_{\overline{D}}$, where $Q_{\overline{D}}$ is a quiver obtained by adding $n-2$ new vertices and $n-2$ arrows to the quiver $Q_{D}$. Additionally, we define a partial order on the set of maximal almost pre-rigid representations, which therefore presents a representation-theoretic interpretation of the type-$\mathbb{D}$ Cambrian lattice determined by $Q_{D}$. Meanwhile, we obtain a representation-theoretic interpretation of the type-$\mathbb{B}$ Cambrian lattices. |
| title | A geometric realization for maximal almost pre-rigid representations over type $\mathbb{D}$ quivers |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2405.03395 |