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

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.