Platonic solids and the McKay correspondence

I teach classes for a linear algebra course and my students had their midterm yesterday. As a break from course content I decided to take a few minutes during our lighthearted class today to tell them about some magic that happens in low dimensional matrix groups. I’ve posted enthusiastically about the McKay correspondence before and noted how it was my introduction to algebraic geometry. This particular part of it is a classical story that I’ve known for a while but never committed to LaTeX until now.

In brief, like so many classifications, the classification of finite subgroups of \text{SL}_2(\mathbb{C}) contains a few infinite families and then a small number of exceptional cases. In this case, the classification is parameterised by the simply-laced Dynkin diagrams, and the exceptionals correspond to the exceptional Dynkin diagrams E_6,E_7,E_8. Where they arise geometrically is a more interesting question, to which the answer is the Platonic solids. It was known to the Greeks that there are five ‘regular’ polyhedra that fall into three dual pairs, and hence produce 3 distinct symmetry groups. It follows by some quick quaternionic algebra that these groups – naturally embedded in \text{SO}_3(\mathbb{R}) – lift to finite subgroups in \text{SU}_2(\mathbb{C})\subset\text{SL}_2(\mathbb{C}), which account for the exceptional Dynkin labels.

Of course there is the undercurrent of algebraic geometry surging through this story, but it’s nice to relive a more elementary view of things on occasion.

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