Here you can see the slides of a talk I’m giving:
• The dodecahedron, the icosahedron and E8, Annual General Meeting of the Hong Kong Mathematical Society, Hong Kong University of Science and Technology.
It’ll take place on 10:50 am Saturday May 20th in Lecture Theatre G. You can see the program for the whole meeting here.
The slides are in the form of webpages, and you can see references and some other information tucked away at the bottom of each page.
In preparing this talk I learned more about the geometric McKay correspondence, which is a correspondence between the simply-laced Dynkin diagrams (also known as ADE Dynkin diagrams) and the finite subgroups of
There are different ways to get your hands on this correspondence, but the geometric way is to resolve the singularity in where is such a finite subgroup. The variety has a singularity at the origin–or more precisely, the point coming from the origin in To make singularities go away, we ‘resolve’ them. And when you take the ‘minimal resolution’ of this variety (a concept I explain here), you get a smooth variety with a map
which is one-to-one except at the origin. The points that map to the origin lie on a bunch of Riemann spheres. There’s one of these spheres for each dot in some Dynkin diagram—and two of these spheres intersect iff their two dots are connected by an edge!
In particular, if is the double cover of the rotational symmetry group of the dodecahedron, the Dynkin diagram we get this way is :
The basic reason is connected to the icosahedron is that the icosahedral group is generated by rotations of orders 2, 3 and 5 while the Dynkin diagram has ‘legs’ of length 2, 3, and 5 if you count right:
In general, whenever you have a triple of natural numbers obeying
you get a finite subgroup of that contains rotations of orders and a simply-laced Dynkin diagram with legs of length The three most exciting cases are:
• : the tetrahedron, and
• : the octahedron, and
• : the icosahedron, and
But the puzzle is this: why does resolving the singular variety gives a smooth variety with a bunch of copies of the Riemann sphere sitting over the singular point at the origin, with these copies intersecting in a pattern given by a Dynkin diagram?
It turns out the best explanation is in here:
• Klaus Lamotke, Regular Solids and Isolated Singularities, Vieweg & Sohn, Braunschweig, 1986.
In a nutshell, you need to start by blowing up at the origin, getting a space containing a copy of on which acts. The space has further singularities coming from the rotations of orders and in . When you resolve these, you get more copies of which intersect in the pattern given by a Dynkin diagram with legs of length and
I would like to understand this better, and more vividly. I want a really clear understanding of the minimal resolution For this I should keep rereading Lamotke’s book, and doing more calculations.
I do, however, have a nice vivid picture of the singular space For that, read my talk! I’m hoping this will lead, someday, to an equally appealing picture of its minimal resolution.