Direct proof of Mckay Correspondence and the representations of finite subgroups of $\mathrm{SO}(4)$

The classic Mckay correspondence gives a connection between finite subgroups of $\mathrm{SU}(2)$ and the simply-laced Dynkin diagrams. In this article, a direct proof is presented. The bipartite structure of the Mckay diagrams is introduced. After that, the similar method can be used on finite subgroups of $\mathrm{SO}(4)$, we get a edges-coloured graph. We finally get some applications.