1. A geometric realization for maximal almost pre-rigid representations over type $\mathbb{D}$ quivers
- Author
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Chen, Jianmin and Zheng, Yiting
- Subjects
Mathematics - Representation Theory - Abstract
We focus on a class of special representations over a type $\mathbb{D}$ quiver $Q_{D}$ with $n$ vertices and directional symmetry, namely, maximal almost pre-rigid representations. By using the equivariant theory of group actions, we give a geometric model for the category of finite dimensional representations over $Q_{D}$ via centrally-symmetric polygon $P(Q_{D})$ with a puncture, and show that the dimension of extension group between indecomposable representations can be interpreted as the crossing number on $P(Q_{D})$. Furthermore, we provide a geometric realization for maximal almost pre-rigid representations over $Q_{D}$. As applications, we prove that each maximal almost pre-rigid representation will determine two or four tilting objects over the path algebra $\Bbbk Q_{\overline{D}}$, where $Q_{\overline{D}}$ is a quiver obtained by adding $n-2$ new vertices and $n-2$ arrows to the quiver $Q_{D}$, and we obtain a representation-theoretic interpretation of the type-$\mathbb{B}$ Cambrian lattices.
- Published
- 2024