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あきみつねざき
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1 1 Slodowy

3 3 1 ADE ADE (0) ADE (1) SL(, C), R 3 (), (3) ADE, II 1 (4) SL(, Z)- (5) F 4, B, M McKay E 6, E 7, E 8

4 4 A n D n E E E

5 5 (4) SL(, Z)- Z =Trq L 0+ L 0 q = e π 1τ 1. ( ). χ k A (1) 1 -, ) 3. Z A (1) 1 - χχ 4. Z SL(, Z)- τ τ 1 ( τ H: ) ) : χ ADE Coxeter

6 6 Cappelli, Itzykson-Zuber, Type k + Z(q, θ, q, θ) A n n +1 n λ=1 χ λ D r 4r r 1 λ=1 χ λ 1 + χ 4r+1 λ + χ r 1, D r+1 4r r λ=1 χ λ 1 + r 1 λ=1 (χ λ χ 4r λ + χ λ χ 4r λ )+ χ r E 6 1 χ 1 + χ 7 + χ 4 + χ 8 + χ 5 + χ 11 E 7 18 χ 1 + χ 17 + χ 5 + χ 13 + χ 7 + χ 11 + χ 9 +(χ 3 + χ 15 ) χ 9 + χ 9 ( χ 3 + χ 15 ) E 8 30 χ 1 + χ 11 + χ 19 + χ 9 + χ 7 + χ 13 + χ 17 + χ 3

7 7 Type Z Coxeter E 6 χ 1 + χ 7 + χ 4 + χ 8 + χ 5 + χ 11 1, 4, 5, 7, 8, 11 E 7 χ 1 + χ 17 + χ 5 + χ 13 + χ 7 + χ 11 + χ 9 +(χ 3 + χ 15 ) χ 9 + χ 9 ( χ 3 + χ 15 ) 1, 5, 7, 9, 11, 13, 17 E 8 χ 1 + χ 11 + χ 19 + χ 9 + χ 7 + χ 13 + χ 17 + χ 3 1, 7, 11, 13, 17, 19, 3, 9

8 8 (5) McKay 3 observation M. (Fisher involution) a, b. 1. ab 1,, 3, 4, 5, 6.. Fisher involution, ab M 9 3. ab : E 8 6 Ẽ

9 9 McKay (1) = (0) McKay ( - ) (1) = (0) + () G-Hilb(C ), G SL() (0) ADE (1) SL(, C) () ) (3) ADE

10 10 1. : (1) = () ). () = (0) (0) ADE (1) SL(, C) () ) (3) ADE

11 11 1. (3) = () Grothendieck, Brieskorn, Slodowy. (1) = (0) (McKay ) 3. McKay (1) + () = (0) + () (Gonzalez-Sprinberg, Verdier) McKay ( - ) (1) = (0) + () G-Hilb(C ), G SL()

12 1 (4) : (Ann. Math. ) (5) : Conway, Lam- - John McKay 003 CRM-Fields-PIMS (Canada McKay Monster McKay observation

13 13 McKay Fischer, Fischer Thompson ( ).. You are reading tea leaves. 78. (5) 1979.

14 McKay Fischer Griess (Fisher-Thompson) (5) - (Griess) 14

15 15 3 McKay - G SL(3) G-Hilb(C 3 ) 1. Smoothness G :. Smoothness 3 McKay (G : - G Bridgeland-King-Reid) 3. G-Hilb G : Craw-Reid 4. G-Hilb(C 3 ) M θ G Craw- 5. G-Hilb G - -

16 16 McKay (1) = (0) McKay ( - ) (1) = (0) + () G-Hilb(C ), G SL() (0) ADE (1) SL(, C) () )

17 17 : 1 G G(D 5 ), 1 σ = G σ, τ ζ ζ6 1 τ =

18 18 e e σ σ τ τ 3 ρ 0 χ ρ 1 χ ρ χ ρ 3 χ ρ 4 χ i i ρ 5 χ i i

19 19 ρ nat : G SL(, C) ρ i ρ ρ nat = ρ 0 + ρ 1 + ρ 3, ρ 0 ρ nat = ρ 1 ρ nat = ρ, ρ 3 ρ nat = ρ + ρ 4 + ρ 5, ρ 4 ρ nat = ρ 3, ρ 5 ρ nat = ρ 3. D 5 ρ 0 ρ ρ 3 ρ 4 ρ ρ 5 : ρ i ρ j ρ i ρ nat = ρ j + ρ 0 ( )

20 0 ( ) σ = G G(D 5 ), 1 G σ, τ ζ ζ6 1 τ = C /G G(D 5 ) F = x 6 + y 6,G = x y,h = xy(x 6 y 6 ) G 4 GF + H =0

21 1 G(D 5 ) G 4 GF + H =0 D 5 X 4 + XY + Z =0 (F, G, H) =(0, 0, 0) (0, 0, 0). C 1 C C 4 C 5 C 3 C 1 C C 3 C 4 6 = C 5 D 5 C i,

22 (0, 0, 0). C 1 C C 4 C 5 C 3 C 1 C C 3 C 4 6 = C 5 D 5 C i :, D 5 : C i C i = ( )

23 3 C /G G

24 4 3 ( ) σ = G G(D 5 ), 1 G σ, τ ζ ζ6 1 τ = C /G G(D 5 ) F = x 6 + y 6,G = x y,h = xy(x 6 y 6 ) G 4 GF + H =0

25 5 G(D 5 ) G 4 GF + H =0 D 5 X 4 + XY + Z =0 (F, G, H) =(0, 0, 0) H/G = (x6 y 6 ) xy xy, x 6 y 6 G G

26 6 3 C /G C /G = {1 G - } : G - : ( G - ) G - Hilb(C ):={ 1 O C - G - } M G - Hilb(C ) 0 I O C M 0( ) I G- G- I F = xy t(x 6 y 6 )

27 7 4 n X n X n n.. n Z ( n = n n r ) Z = n 1 P 1 + n P + + n r P r (P i P j ) 4 X =C P i : x = α i Z I Z = f Z C[x] Z f Z (x). f Z (x) =(x α 1 ) n 1 (x α ) n (x α r ) n r = x n + a 1 x n 1 + a x n + + a n 1 x + a n

28 8 Z n I Z C[x] n., Z= n f Z = x n. X =C Hilb n (C) = {X n } = {x n,x n 1} n 1 = xn + a j x j ; a j C =Cn j=0

29 9 5 X C Z = n 1 P n r P r, ( ), P i P j Z X }{{} X/( ) =X (n) n X (n) = X }{{} X/S n n. X (n). X (n) Hilb n (C )

30 30 X [n] =Hilb n (C ). X [n] I C[x, y]; I = C[x, y] n I C[x, y]; I = xi I,yI I, dim C[x, y]/i = n 1 (Fogarty 1968) X =C. X [n] X (n).

31 31 π : X [n] X (n) P 1,,P r Z π : Z n 1 P n r P r. n i = Z P i.

32 3 (Fogarty 1968)( ) X [N] = Hilb N (C ) X (N) ( ) G SL(), N = G G π G- :(X [N] ) G- (X (N) ) G- Fogarty (X [N] ) G-. G- =

33 33 5 G- 3 (X (N) ) G- =C /G. 4 G - Hilb(C ):=(X [N] ) G- G - Hilb(C ) G-. I G - Hilb(C ) C[x, y]/i =C[G] 5 ( ) G - Hilb(C ) ( ) C /G.

34 34 6 ( ) G - Hilb(C ) ( ) C /G. ( ) (1 1 ) (1 1 ) ( ) G ( )

35 35 A n... D n... D 5 E 6 E 7 E 8

36 36 A n... D n... D 5 E 6 E 7 E 8

37 37 A n... D n... D 5 E 6 E 7 E 8

38 38 A n... D n... D 5 E 6 E 7 E 8

39 39 A n... D n... D 5 E 6 E 7 E 8

40 40 A n... D n... D 5 E 6 E 7 E 8

41 41 A n... D n... D 5 E 6 E 7 E 8

42 A n... (n vertices) 1 1 D n D 5 E 6 1 E 7 3 E (n vertices)

43 43 6 D 5 G - Hilb(C ) C /G. : E = {I G - Hilb(X); I m} m =(x, y)c[x, y]. I E V (I) =I/(mI + n). I. n =(F, G, H)C[x, y]. V (I) G- G

44 44 E E 1 V (ρ 1 )={xy},v 6 (ρ 1 )={x 6 y 6 } I 1 (s) ={xy + s(x 6 y 6 )} + n I 1 (s) C[x, y], xy + s(x 6 y 6 ) n =(F, G, H). dim C[x, y]/i 1 (s) =1, I 1 (s) G - Hilb(C ) ( s C)

45 45 s : E I I n =(F, G, H) C[x, y]/n C[x, y]/i E Grassmann(C[x, y]/n, 11 I 1 (s) (s C). I 1 ( )/n = lim s I 1 (s)/n I 1 ( ) =

46 46 I 1 (s) ={xy + s(x 6 y 6 )} + n (s 0) I 1 (s) ={ 1 s xy +(x6 y 6 )} + n (s 0) s = 1 s =0 I 1 ( ) ={x 6 y 6 } + n I 1 ( ) / X [1]. : I 1 ( ) ={x 6 y 6 } + {x y, xy } + n n s 0 x y, xy I 1 (s) V 3 (ρ )={x y, xy } = {x, y} {xy} = ρ nat V (ρ 1 ) ρ = ρ nat ρ 1

47 47 E E 1 :={I E; G - V (I) ρ 1 } = {I G - Hilb; m I,ρ 1 V (I)} E := {I E; V (I) ρ }. E 1 = {I 1 (s); s C} I 1 ( ) P 1

48 48 V 3 (ρ )={x y, xy }, V 5 (ρ 1 )={ y 5,x 5 } t 0 I (t) =(x y ty 5,xy + tx 5 )+n V (I (t)) ρ. E = {I E; V (I) ρ } P 1 t 0 I 1 ( ) I (0) = (x y, xy )+n

49 49 t 0 I (t) =(x y ty 5,xy + tx 5 )+n, I (0) = (x y, xy )+(x 6 y 6 )+n = I 1 ( ) V 6 (ρ 1 ) = {x 6 y 6 } = {x, y} { y 5,x 5 } mod n = ρ nat V 5 (ρ ) mod n +(x y, xy ) ρ 1 + = ρ nat ρ D 5 ρ 0 ρ ρ 3 ρ 4 ρ ρ 5

50 50 I 1 (s) ={(1/s)(xy) +(x 6 y 6 )+n I 1 ( ) =(x y, xy ) +(x 6 y 6 )+n V 3 (ρ )={x y, xy } = {x, y} {xy} = ρ nat V (ρ 1 ) ρ = ρ nat ρ 1 I (t) =(x y + t( y 5 ),xy + tx 5 )+n I (0) = (x y, xy )+(x 6 y 6 ) + n V 6 (ρ 1 ) {x 6 y 6 } {x, y} { y 5,x 5 } ρ nat V 5 (ρ ) mod n +(x y, xy ) ρ 1 + = ρ nat ρ (1/s) =0 t =0

51 51 E(ρ 1 ) E(ρ ) E(ρ 1 ) I 1 ( ) E(ρ ) I 1 (s) (s C) I (t) (t C ) I (0) V (I 1 (s)) ρ 1, V (I (t)) ρ, V (I 1 ( )) = V (I (0)) ρ 1 ρ s C, t C(s 0) V (I) =I/(mI + n) I

52 5 D 5 ρ 0 ρ ρ 3 ρ 4 ρ ρ 5 ρ ρ nat = ρ 0 + ρ 1 + ρ 3 ρ 1 ρ nat = ρ

53 53 m V m (ρ) 1 {x, y} ρ {xy} {x,y } ρ 1 + ρ 3 3 {x y, xy } {x 3 ± iy 3 } ρ + ρ ρ 5 4 {y 4,x 4 } {x 3 y, xy 3 } ρ 3 5 {y 5, x 5 } {xy(x 3 ± ( iy 3 ))} ρ + ρ ρ 5 6 {x 6 y 6 } {x 5 y, xy 5 } ρ 1 + ρ 3 7 {xy 6,x 6 y} ρ C[x, y]/n =C[x, y]/(f, G, H) I G - Hilb(C ) C[x, y]/i =C[G]

54 ρ ρ 1 + ρ 3 ρ + ρ 4 + ρ 5 ρ ρ ρ 1 + ρ 3 ρ + ρ 4 + ρ 5 Quiver V 1 = {x, } V 1 V (ρ 1 )=V 3 (ρ ), V 1 V 3 (ρ ) V 4 (ρ 3 ), V 1 V 3 (ρ 4 ) V 4 (ρ 3 ), V 1 V 3 (ρ 5 ) V 4 (ρ 3 ), V 1 V 4 (ρ 3 ) V 5 (ρ ) V 5 (ρ 4 ) V 5 (ρ 5 ), V 1 V 5 (ρ ) V 6 (ρ 1 ), V 1 V 5 (ρ 4 ) 0, V 1 V 5 (ρ 5 ) 0

55 ρ ρ 1 + ρ 3 ρ + ρ 4 + ρ 5 ρ ρ ρ 1 + ρ 3 ρ + ρ 4 + ρ 5 Quiver 3: V 1 ρ 1 = ρ, V 1 = {x, y} 4: V 1 ρ = ρ 3, V 1 ρ 4 = ρ 3, V 1 ρ 5 = ρ 3 5: V 1 ρ 3 ρ ρ 4 ρ 5 6: V 1 ρ ρ 1, V 1 ρ 4 0, V 1 ρ 5 0 C[x, y]/i =C[G] I

56 56 D 5 Quiver V (ρ 1 ) V 6 (ρ 1 ) V 3 (ρ ) V 5 (ρ ) V 4 (ρ 3 ) V 3 (ρ 4 ) V 5 (ρ 4 ) V 3 (ρ 5 ) V 5 (ρ 5 )

57 57 D 5 V (ρ 1 ) V 6 (ρ 1 ) V 3 (ρ ) V 5 (ρ ) V 4 (ρ 3 ) V 3 (ρ 4 ) V 5 (ρ 4 ) V 3 (ρ 5 ) V 5 (ρ 5 )

58 58 V (ρ 1 ) V 6 (ρ 1 ) V 3 (ρ ) V 5 (ρ ) V 4 (ρ 3 ) V 3 (ρ 4 ) V 5 (ρ 4 ) V 3 (ρ 5 ) V 5 (ρ 5 )

59 59 7 E 6 G(E 6 ) SL(, C) (Binary Tetrahedral Group) σ = G(E 6 )= σ, τ, µ 4 i, 0, τ = 0, 1, µ = 1 0, i 1, 0 where ɛ = e πi/8. ɛ7,ɛ 7, ɛ 5, ɛ G(D 4 ):= σ, τ, normal in G(E 6 ) 1 G(D 4 ) G(E 6 ) Z/3Z 1. C /G(D 4 ) 3:1 C /G(E 6 )

60 60 E 6 ) 1 1 τ µ µ µ 4 µ 5 ( ) ρ ρ ρ ρ 0 ω ω ω ω ρ ω ω ω ω ρ 0 ω ω ω ω ρ ω ω ω ω

61 61 E 6 m V m 1 ρ ρ 3 3 ρ + ρ 4 ρ 1 + ρ 1 + ρ 3 5 ρ + ρ + ρ 6 ρ 3 7 ρ + ρ + ρ 8 ρ 1 + ρ 1 + ρ 3 9 ρ + ρ 10 ρ 3 11 ρ

62 6 V 5 (ρ ) V 7 (ρ ) V 4 (ρ 1 ) V 8 (ρ 1 ) V 5 (ρ ) V 7 (ρ ) V 6 (ρ 3 ) V 5 (ρ ) V 7 (ρ ) V 4 (ρ 1 ) V 8 (ρ 1 ) E 6

63 63 V 5 (ρ ) V 7 (ρ ) V 4 (ρ 1 ) V 8 (ρ 1 ) V 5 (ρ ) V 7 (ρ ) V 6 (ρ 3 ) V 5 (ρ ) V 7 (ρ ) V 4 (ρ 1 ) V 8 (ρ 1 ) E 6

64 64 A n... D n... D 5 E 6 E 7 E 8

65 65 A n... D n... D 5 E 6 E 7 E 8

66 66 A n... D n... D 5 E 6 E 7 E 8

67 67 A n... D n... D 5 E 6 E 7 E 8

68 68 A n... D n... D 5 E 6 E 7 E 8

69 69 A n... D n... D 5 E 6 E 7 E 8

70 70 A n... D n... D 5 E 6 E 7 E 8

71 71 8 I G - Hilb(C ) I E G - Hilb(C )

72 7 E I n V (I) :=I/(mI + n). V (I) G ρ, ρ E : E(ρ) :={I E; V (I) ρ} P (ρ, ρ ):={I E; V (I) ρ ρ }

73 73 7 ( ) G SL(, C). (1) G-Hilb(C ) C /G. () G ρ E(ρ) P 1, ρ E(ρ) (3) ρ nat Quiver E(ρ) P (ρ, ρ ).

74 74

L P y P y + ɛ, ɛ y P y I P y,, y P y + I P y, 3 ŷ β 0 β y β 0 β y β β 0, β y x x, x,, x, y y, y,, y x x y y x x, y y, x x y y {}}{,,, / / L P / / y, P

005 5 6 y β + ɛ {x, x,, x p } y, {x, x,, x p }, β, ɛ E ɛ 0 V ɛ σ I 3 rak p 4 ɛ i N 0, σ ɛ ɛ y β y β y y β y + β β, ɛ β y + β 0, β y β y ɛ ɛ β ɛ y β mi L y y ŷ β y β y β β L P y P y + ɛ, ɛ y P y I P y,,