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

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