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