McKay correspondence

TL;DR

The notes examine the McKay correspondence for finite groups $G\subset SL(n,\mathbb{C})$, linking irreducible representations and conjugacy classes to the cohomology and homology of a crepant resolution $Y\to X=\mathbb{C}^n/G$, and discuss how these bijections interact with the McKay quiver and product structures. They develop the tautological $G$-sheaves $\mathcal{F}_\rho$ and propose a Main Conjecture that these sheaves form a $\mathbb{Z}$-basis of $K_0(\mathrm{Coh}\,Y)$ and, via Chern characters, a $\mathbb{Z}$-basis of $H^*(Y,J)$ in crepant cases, with extensions to derived categories. The text surveys Ito–Reid’s direct correspondence in dimension three, and Nakamura’s toric/cluster-based proof that $\operatorname{G{-}Hilb}\mathbb{C}^3$ furnishes a crepant resolution for diagonal $G\subset SL(3)$, using a honeycomb/McKay quiver framework and explicit equations for $G$-clusters. Through detailed examples (e.g., $A_n$-type, $1/r(1,q)$ surfaces, and toric cases), it illustrates how ratios of covariants yield natural linear systems that generate cohomology, and outlines a Beilinson-type program to relate diagonal data to derived-category structures. The discussion highlights connections to mirror symmetry, the role of junior elements and age, and a program to unify geometric resolutions with representation-theoretic data in higher dimensions. Overall, the work points toward a coherent, tautological perspective on McKay-type correspondences, with concrete constructive proofs in key cases and broad implications for $G$-Hilb and $G$-mirror symmetry.

Abstract

This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's formula, how the McKay correspondence for finite subgroups of SL(n,C) relates to mirror symmetry. The main aim is to give numerical examples of how the 2 McKay correspondences (1) representations of G <--> cohomology of resolution (2) conjugacy classes of G <--> homology must work, and to restate my 1992 Conjecture as a tautology, like cohomology or K-theory of projective space. Another aim is to give an introduction to Nakamura's results on the Hilbert scheme of G-clusters, following his preprints and his many helpful explanations. This is partly based on joint work with Y. Ito, and has benefited from encouragement and invaluable suggestions of S. Mukai.

Figures (13)

• Figure 1: $E_0$ and $E_r$ are the image of the $x$ and $y$ axes
• Figure 2: Triangulation of $\Delta_{\text{junior}}$ in Example \ref{['ex:max']}
• Figure 3: The resolution corresponding to the triangulation of Figure \ref{['fig:max1']}
• Figure 4: Two affine pieces near the hexagon at (3,1,1)
• Figure 5: McKay correspondence for Example \ref{['ex:max']}

Theorems & Definitions (22)

• Conjecture 1.1: since 1992
• Conjecture 1.2: "physicists' Euler number conjecture"
• Example 2.1
• Example 2.2
• Conjecture 4.1: Main conjecture
• Remark 4.2
• Example 4.3
• Remark 4.4
• Definition 5.1
• Remark 5.2