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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/2405.03395 |
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Table of 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.