mckay correspondence iku nakamura

MCKAY CORRESPONDENCE 31 1 1 1 1A n ✈ ✈ ✈ ... ✈ ✈ (n vertices)✈1 2 2 2 1D n ✈ ✈ ✈ ... ✈ ❅ ❅✈ 1✈2E 6 ✈ ✈ ✈ ✈ ✈1 2 3 2 1✈2E 7 ✈ ✈ ✈ ✈ ✈ ✈2 3 4 3 2 1✈3E 8 ✈ ✈ ✈ ✈ ✈ ✈ ✈2 4 6 5 4 3 2(n vertices)Figure 1. The Dynkin diagrams ADEType G name |G| h (d 1 ,d 2 ,d 3 )A n Z n+1 cyclic n +1 n +1 (2,n+1,n+1)D n D n−2 binary dihedral 4(n − 2) 2n − 2 (4, 2n − 4, 2n − 2)E 6 T binary tetrahedral 24 12 (6, 8, 12)E 7 O binary octahedral 48 18 (8, 12, 18)E 8 I binary icosahedral 120 30 (12, 20, 30)Table 1. Finite subgroups of SL(2, C)( , ) SING can be expressed by a finite graph with simple edges. We rephrasethis as follows: we associate a vertex v(E ′ ) to any irreducible component E ′of E, and join two vertices v(E ′ ) and v(E ′′ ) if and only if (E ′ E ′′ ) SING =1.Thus we have a finite graph with simple edges. We call this graph the dualgraph of E, and denote it by Γ SING (S) or Γ(Irr E).There exists a unique divisor E fund , called the fundamental cycle of X,which is the minimal nonzero effective divisor such that E fund E i ≤ 0 for alli. Let E fund := ∑ ri=1 mSING i E i and E 0 := −E fund . For the simple singularitieswe have E 0 E i = 0 or 1 for any E i ∈ Irr E (except for the case A 1 , whenE 0 E 1 = 2). Thus we can draw a new graph ˜Γ SING by adding the vertex v(E 0 )to Γ SING (S). By abuse of notation we denote Irr E∪{E 0 } by Irr ∗ E. Also for agiven finite subgroup G of SL(2, C), we have a quotient singularity (A 2 /G, 0),which is one of simple singularities so that we have a Dynkin diagram as a dual