Finding Euler Characteristics of Hilbert Schemes using Colored Young Diagrams
The Hilbert schemes of the singular space and of the orbifold are two structures that contain geometric data about group actions on a polynomial ring. Our goal is to understand this geometry by finding the Euler Characteristics of these spaces. The problem is equivalent to counting Young diagrams that are based on the group action. For the singular case, we count all zero generated Young diagrams that contain a certain number of 0 colored squares, and we prove a theorem greatly reducing the problem, sometimes into already solved cases. For the Hilbert scheme of the orbifold, we count all Young diagrams with a given coloring, and we develop a procedure to obtain the desired generating function, as well as closed form generating functions for special cases. We also explore the method of vertex operator algebras.