mckay correspondence iku nakamura

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10 IKU NAKAMURALet S = C[x, y]. For any positive integer k let S k be the subspace of homogeneouspolynomials of degree k in S. We say that a G-submodule W ofm/n is homogeneous of degree k if it is generated over C by homogeneouspolynomials of degree k. The G-module m/n splits as a direct sum of irreduciblehomogeneous G-modules. If W is a direct sum of homogeneousG-submodules, then we denote the homogeneous part of W of degree k byS k (W ). For any G-module W in some S k (m/n), we write S j · W for the G-submodule of S k+j (m/n) generated over C by the products of S j (m/n) andW . We denote by W [ρ] the ρ factor of W , that is, the sum of all the copiesof ρ in W ; and similarly, we denote by [W : ρ] the multiplicity of ρ ∈ Irr G ina G-module W .Definition 3.5. The quotient algebra S/n is called the coinvariant algebraof G, denoted by Coinv(G) (or denoted by Coinv(the type of S G ) with thenotation of Table 1). Let h be the Coxeter number of the simple singularityS G . Then we define the very positive part S † := S † (m/n) of S/n to be∑S † :=S k (m/n)[ρ].k> h 2 +d(ρ),ρ∈Irr GWe also define the McKay quiver of G byS McKay (G) = ∑S h2 ±d(ρ)(m/n)[ρ]+S† /S † ,ρ∈Irr Gwhich we also denote by S McKay (the type of S G ). We also define V k (ρ) =S k (m/n)[ρ] and V (ρ) =S McKay (G)[ρ].The following Lemma 3.6 and Lemma 3.7 describe the structure of theMcKay quiver S McKay (G). See [IN99] for the proofs.Lemma 3.6. Let ϕ i be three generators of G-invariants, d i = deg ϕ i and hthe Coxeter number of the simple singularity S G . Then d 1 + d 2 = d 3 +2 andd 3 = h, and moreover S k (m/n) =0for k ≥ d 3 .Lemma 3.7. Assume that G is not cyclic. Let h be the Coxeter number ofthe simple singularity S G . Then as G-modules, we have(1) m/n = ∑ ρ∈Irr G2(deg ρ)ρ;(2) S McKay (G) ≃ ∑ ρ∈Irr G 2ρ;(3) V h−d(ρ)(ρ) ≃ V h +d(ρ)(ρ) =ρ if d(ρ) ≥ 1 and V h (ρ) =ρ ⊕2 if d(ρ) =0;2 2 2(4) S h−k(m/n) ≃ S h +k(m/n) for any k.2 2Using this notation we see by Lemma 3.7, (2) that V (ρ) =V h−d(ρ)(ρ) +2V h+d(ρ)(ρ) ≃ ρ⊕2 and that the subset E(ρ) isP(V (ρ)), the set of all nontrivial2G-submodules of ρ ⊕2 . This is isomorphic to the projective line, or a smoothrational curve by Schur’s lemma. This proves Theorem 3.4, (2).
MCKAY CORRESPONDENCE 113.8. The ideals n X and π ∗ m. Since (A 2 /G) × ( 2 /G) X ≃ X, X is a closedsubscheme of (A 2 /G) × X, which is defined by an ideal I X of O 2 /G ⊗ O X .Let n X be the ideal of O 2 ⊗ O X generated by I X . Let Z univ be the universalsubscheme of A 2 × X, and I univ the ideal sheaf of O 2 ×X defining Z univ . Sincewe have a commutative diagrampr 2Z univ −−−→ X⏐ pr 1⏐↓⏐π↓A 2φ−−−→ A 2 /G,the morphism φ × id X : A 2 × X → A 2 /G × X sends Z univ into (A 2 /G) × ( 2 /G)X ≃ X. This implies that n X = I X O 2 ⊗ O X ⊂ I univ . Now we defineV := V (I univ )=I univ /mI univ + n X .Let m be the maximal ideal of O 2 /G of the unique singular point. Wenote n =Γ(A 2 ,φ ∗ m). We also note π ∗ (m) ∩ I univ = {0}. In fact, supposeπ ∗ (F ) ∈ I univ . Then π ∗ (F )=0onZ univ . Since Z univ is surjective over A 2 /G,F = 0. It follows π ∗ (m) ∩ I univ = {0}.Since π ∗ m := mO X is the defining ideal of E fund by [Artin66], we see thatV is a finite O Efund -module. See section 6 for the proof. However we prove alittle strongerTheorem 3.9. The O 2 × X-module V is a finite O E -module; as an O 2 × X-module with G-action, we have an isomorphismV≃ ⊕ρ ⊗ O E(ρ) (−1).ρ∈Irr GIn section 6 we will prove Theorem 3.9. In order to prepare for the proofof Theorem 3.9, we recall in section 4 the proof of Theorem 3.4 and relatedconstructions in the case of D 5 in full detail. In the cases D n (n ≠ 5) and E 6we give only a sketch of proofs of Theorem 3.9. Though we give almost noproof for E 7 because we need more notation, we remark that we can prove itin the same manner. It remains to check E 8 .4. The Simple Singularity D 5In this section we explain the case of D 5 in full detail.4.1. The binary dihedral group D 3 . The simple singularity D 5 is thequotient singularity of A 2 by the binary dihedral group G := D 3 of order 12,which is generated by σ and τ:σ =( )ɛ 00 ɛ −1 , τ =( )0 1,−1 0where ɛ := e 2πi/6 . We have σ 6 = τ 4 =1,σ 3 = τ 2 and τστ −1 = σ −1 . Thering of G-invariants in C[x, y] is generated by three elements A 6 := x 6 + y 6 ,
Page 1: MCKAY CORRESPONDENCEIKU NAKAMURAAbs
Page 5 and 6: MCKAY CORRESPONDENCE 5ρ Tr ρ 1 σ
Page 7: MCKAY CORRESPONDENCE 7The point set
Page 12 and 13: 12 IKU NAKAMURAA 8 := xy(x 6 − y
Page 14 and 15: 14 IKU NAKAMURAby an edge. To descr
Page 16: 16 IKU NAKAMURAthe deformation to a
Page 24: 24 IKU NAKAMURA{0}⊕V 6 (ρ 1 )✲
Page 27 and 28: MCKAY CORRESPONDENCE 27❣✻❝❣
Page 30 and 31: 30 IKU NAKAMURASince T [I] (E(ρ 2
Page 32 and 33: 32 IKU NAKAMURAThen we see p 1 B 1
Page 34 and 35: 34 IKU NAKAMURASince V is O E(ρ) -
Page 36: 36 IKU NAKAMURA6.4.2. Let I ∈ E(

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