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nLab > Latest Changes: McKay correspondence

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 17th 2018

    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 8E_8 and the icosahedron.

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2018

    I added mention of Gonzalez-Sprinberg and J.-L. Verdier’s K-theoretic interpretation as discussed in the ’The McKay correspondence as an equivalence of derived categories’.

    diff, v6, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 9th 2018

    If the comparison is between equivariant K-theory of ℂ 2\mathbb{C}^2 and ordinary K-theory of ℂ 2/G\mathbb{C}^2/G, is that an Elmendorf-theorem situation?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2018

    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).

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 9th 2018

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2018

    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.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2019

    finally added pointer to the original reference

    • John McKay, Graphs, singularities, and finite groups Proc. Symp. Pure Math. Vol. 37. No. 183. 1980

    diff, v22, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 23rd 2019
    • (edited Jan 23rd 2019)

    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.

    diff, v25, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2020

    added pointer to today’s

    • Duiliu-Emanuel Diaconescu, Mauro Porta, Francesco Sala, McKay correspondence, cohomological Hall algebras and categorification (arXiv:2004.13685)

    diff, v26, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2020

    made some minor cosmetic edits to the text

    diff, v27, current

1 to 10 of 10

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