References & Citations
Mathematics > Representation Theory
Title: A geometric realization for maximal almost pre-rigid representations over type $\mathbb{D}$ quivers
(Submitted on 6 May 2024 (v1), last revised 7 May 2024 (this version, v2))
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 an application, we illustrate their general form and 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 $Q_{D}$.
Submission history
From: Yiting Zheng [view email][v1] Mon, 6 May 2024 12:03:58 GMT (40kb)
[v2] Tue, 7 May 2024 15:45:24 GMT (40kb)
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