A geometric realization for maximal almost pre-rigid representations over type $\mathbb{D}$ quivers

TL;DR

The paper constructs a geometric model for the category of finite-dimensional representations of a type $\mathbb{D}$ quiver $Q_{D}$ with symmetry by leveraging an equivariant setup from a type $\mathbb{A}$ quiver $Q_{A}$ with a $\mathbb{Z}_2$-action. It introduces a punctured polygon $P(Q_{D})$ and a category of tagged line segments $\mathcal{C}_{D}$ to realize $\Bbbk Q_{D}$-mod as $\mathcal{C}_{D}$, with extension spaces measured by crossing numbers of tagged segments. Maximal almost pre-rigid representations over $Q_{D}$ correspond bijectively to tagged triangulations of $P(Q_{D})$, and their endomorphism algebras are tilted algebras of type $Q_{\overline{D}}$, with a cluster-category interpretation via a trivial extension. The work yields representation-theoretic interpretations of type-$\mathbb{D}$ Cambrian lattices (and type-$\mathbb{B}$ variants) in terms of tagged triangulations and Ext-geometry, and enumerates the objects by the generalized Catalan number, linking combinatorics with tilting theory in a geometric framework.

Abstract

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.

Figures (12)

• Figure 1: $\Sigma_{\rightarrow}(M)$(left) and $\mathscr{R}_{\rightarrow}(M)$(right) of $M$
• Figure 2: The underlying graph of $Q_{A_{2n-3}}$
• Figure 3: The underlying graph of $Q_{D_{n}}$
• Figure 4: The quiver $Q_{A_{7}}$ and the polygon $P(Q_{A_{7}})$
• Figure 5: The quiver $Q_{D_{5}}$ and the polygon $P(Q_{D_{5}})$

Theorems & Definitions (77)

• Theorem A: Theorem \ref{['Thm:A']}
• Theorem B: Theorem \ref{['E and e']}
• Theorem C: Theorem \ref{['Thm 1']}
• Theorem D: Theorem \ref{['Thm:D']}
• Theorem E: Theorem \ref{['Thm:F']}
• Remark 2.1
• Proposition 2.2
• Remark 3.1
• Proposition 3.2
• proof