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I added references to John Baez’s two blog posts on The Geometric McKay Correspondence, Part I, Part II.
I hadn’t realised the length of legs in the Dynkin diagrams corresponds to the stabilizer order on vertices, edges, faces in the corresponding Platonic solid. So 2,3,5 for $E_8$ and the icosahedron.
If the comparison is between equivariant K-theory of $\mathbb{C}^2$ and ordinary K-theory of $\mathbb{C}^2/G$, is that an Elmendorf-theorem situation?
Haven’t looked yet at the article that you are looking at, but Elmendorf’s theorem is about restricting to fixed points, not about passing to quotients (not manifestly and directly, at least).
I was drawing on the Bridgeland et al. article for that K-theoretic description, which they propose to generalise to
G-equivariant K theory of M [=X/G] and the ordinary K theory of a crepant resolution Y of X,
But, as you say, I guess that’s not about fixed points.
Thanks for the pointer. Interesting.
But, yes, that old trick with those du Val resolutions is presently not related, beyond the general context, to the perspective through Elmendorf’s theorem. I wish I knew how to make an tighter connection between the two.
finally added pointer to the original reference
I have expanded and re-organized:
First of all I brought in the paragraphs that I had yesterday typed into McKay quiver to finally state the original idea of the correspondence, as actually given by McKay.
Then I aligned that more systematically with the other two perspectives on (refinements of) the correspondence, that via equivariant K-theory and that via Seiberg-Witten theory.
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