A geometric realization for maximal almost pre-rigid representations over type D quivers

We focus on a class of special representations over a type D quiver Q₃ 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 Q₃ via centrally-symmetric polygon P (Q₃) with a puncture, and show that the dimension of extension group between indecomposable representations can be interpreted as the crossing number on P (Q₃). Furthermore, we provide a geometric realization for maximal almost pre-rigid representations over Q₃. 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 Q₃, where Q₃ is a quiver obtained by adding n-2 new vertices and n-2 arrows to the quiver Q₃.