Îã³°·¿¤Î¥·¥å¡¼¥Ù¥ë¥È¥«¥êto=1=¡á=1=¥ë¥�¥å¥é¥¹

Size: px

Start display at page:

Download "Îã³°·¿¤Î¥·¥å¡¼¥Ù¥ë¥È¥«¥êto=1=¡á=1=¥ë¥�¥å¥é¥¹"

うのすけさわまつ
4 years ago
Views:

2 R 3

3 R 3 ( wikipedia )

5 (Schubert, 19 ) (= )(Ehresmann, 20 ) (Chevalley, 20 ) G/P: ( : ) W : ( : ) X w : W X w W

7 G: B G: Borel P B: G/P: 1 C n ( ) Fl n := {0 V 1 V 2 V n = C n } G = GL n(c) B Fl n = GL n(c)/b: complete flag variety 2 C n m Gr(m, n) GL n (C) «Am P m = 0 A n m Gr(m, n) = GL n(c)/p m:

8 Fact T B G: n = dim T : t: T s 1,...,s n GL(t ): W = s 1,...,s n : l(w) Z 0 : P P {1, 2,...,n} (B ) W P = s i i P : P W P := {w W i / P, l(ws i ) > l(w)} = W/W P P m : P = {1, 2,..., ˆm,...,n} G = GL n, P = P m W = S n, W P = S m S n m W P = {w S n w(1) < < w(m), w(m + 1) < < w(n)}

9 G/B GL n/b SO 2n+1 (C)/B Sp 2n (C)/B SO 2n (C)/B (= 2l(w 0 )) n(n 1) 2n 2 2n 2 2n(n 1) (= #W ) n! 2 n n! 2 n n! 2 n 1 n! G/B G 2 /B F 4 /B E 6 /B E 7/B E 8 /B (= 2l(w 0 )) (= #W )

10 T G/P W P {wp/p}, w W P B X w := BwP/P G = GL n, P = P m w W Pm X w = {V C n dim(v C w(j) ) j} Gr(m, n)

11 : Fl 4 ( ) w = S w = s 2 s 1 s 3 s 2, l(w) = w = Bw = BwB/B = dim C BwB/B = # = l(w) = 4

12 Bruhat W (strong) Bruhat : w v v = [i 1,...i k ] subword [i n1,...,i nj ] = w W(GL 4 (C)) = S 4 Bruhat GL 4 (C)/B W P2 W : GL 4 (C)/P 2 = Gr(2, 4) {v W v [2, 1, 3, 2]}: X [2,1,3,2]

13 Bruhat W (strong) Bruhat : w v v = [i 1,...i k ] subword [i n1,...,i nj ] = w W(GL 4 (C)) = S 4 Bruhat GL 4 (C)/B W P2 W : GL 4 (C)/P 2 = Gr(2, 4) {v W v [2, 1, 3, 2]}: X [2,1,3,2]

15 dim X w = 2l(w) w 0 W X w0 = G/B w v X w X v X w singularity w H (G/P; Z) = w W P Z[X w ] σ w := [X w ] H (G/P; Z) = w W P Zσ w σ w H 2l(w) (G/P; Z) H (G/P; Z) torsion-free H odd (G/P; Z) = 0 σ u σ v = P w cw u,vσ w cu,v structure w constant ( ) X w singularity c w u,v c w u,v 0

16 structure constant Littlewood-Richardson rule Chevalley formula GKM type formula (localization formula) Duan s formula

17 Borel H (G; Z) torsion prime G GL n (C) SO n (C) Sp 2n (C) G 2 F 4, E 6, E 7 E 8 p 2 2 2, 3 2, 3, 5 R := Z[p 1 ] p : torsion primes (Borel) H (G/P; R) = H (BT; R) W P (H + (BT; R) W ) (H (BT; Z) = Z[x 1,...,x n ]) H (BT; R) W P : P (H + (BT; R) W ): G G = GL n(c), Sp 2n (C) R = Z H (G/B; R) 2 {σ [1],..., σ [n] } W {σ [1],..., σ [n] }

18 : H (Gr(m, n); Z) H (Gr(m, n); Z) = Z[x 1,...,x n ] Sm Sn m (Z + [x 1,...,x n ] Sn ) Z[c 1,...,c m, c 1 =,...,c n m] (1 + c 1 (x 1,...,x n ) + + c n (x 1,...,x n )) c i := c i (x 1,...,x m ), c i := c i (x m+1,...,x n ) ) Giambelli (σ w ): σ w = det i,j (c w(m+1 i) 2i+j σ[2] 4 = (c 1 )4 2(c 2 )2 = 2σ [2,1,3,2] ( X w ): [i1,i 2,...,i k ] = i1 ik : Z[c 1,...,c m, c 1,...,c n m] i (f) := f(x 1,...,x i, x i+1,...,x n ) f(x 1,...,x i+1, x i,...,x n ) x i+1 x i [2,1,3,2] (c 1 )4 = [2,1,3,2] (x 3 + x 4 ) 4 = 2

19 H (Gr(2, 4)) R 3 C & (ax + by + cz = d ax + by + cz dw = 0) R 3 C 4 = {V 2 C 4 dim(v 2 C 2 ) 1} = X = X A ( ) s σ [2] R 3 Y := X X X X #Y = Gr(2,4) σ 4 [2] = 2

20 : Giambelli : : (Bruhat ) transition : Giambelli transition ( ) ( )

22 1 {σ wj } π : Z[σ w1,...,σ wk ] H (G/P; Z) 2 ker π = (ρ 1,...,ρ h ) H (G/P; Z) = Z[σ w 1,...,σ wk ] (ρ 1,...,ρ h ) 3 w W σ w {σ wk } σ w = f w (σ w1,...,σ wk ) Z[σ w 1,...,σ wk ] (ρ 1,...,ρ h )

23 : H (G/P; Z) = Zσ w Zσ w = H (G/B; Z) w W P w W σ w σ w H (G/B; Z) : rank(h (E 8 /B)) = W = ! : rank(h (E 8 /P 2 )) = W P2 = 17280

24 ( ) P/B ι G/B p G/P P/B, G/B, G/P Serre E 2 -collapse H (G/P; Z) = H (G/B; Z) W P { Z[σ vi ] H (G/P; Z) Z[ι (σ ui )] H (P/B; Z) Z[σ vi,σ ui ] H (G/B; Z) H (P/B; Z) & H (G/P; Z)

25 A C α α 1 α 2 α 1 α 2 α 3 α 1 α 3 α 4 α 5 α 6 α n 4 α 2 { GL n (C)/B G/B p G/P 2 (G = G 2, E n ) ( ) Sp 2n (C)/B G/B p G/P 1 (G = F 4 ) P m : {1, 2,..., ˆm,...,n} H (GL n /B; Z), H (Sp 2n /B; Z) Borel

26 E t 1 = σ [1] + σ [2], t 2 = σ [1] + σ [2] σ [3], t 3 = σ [2] + σ [3] σ [4], t 4 =σ [4] σ [5], t i = σ [i] σ [i+1],...,t n = σ [n], ( {t i } σ [n] W P 2 - ) H (BT; Z) = σ [1],...,σ [n] = σ [2], t 1,...,t n W P 2 = s1, ŝ 2,...,s n, W = W P 2, s2 W P 2 {t1,...,t n }: W P 2 σ[2] : c i := c i (t 1,...,t n : i ( ) H + (BT; Z) W P 2 = ( ) σ [2], c 2,...,c n H (P 2 /B; Z) = H (GL n (C)/B; Z) = H (BT; Z) ( H+ (BT; Z) W P) = Z[σ [2], t 1,...,t n ] ( σ[2], c 2,...,c n )

27 E GL n (C)/B ι E n /B p E n /P 2

28 E GL n (C)/B ι E n /B p E n /P 2 H (GL n (C)/B; Z) = Z[σ [2], t 1,...,t n ] (σ [2], c 2,...,c n )

29 E GL n (C)/B ι E n /B p E n /P 2 H (GL n (C)/B; Z) = Z[σ [2], t 1,...,t n ] (σ [2], c 2,...,c n ) H (E n /P 2 ; Z) = H (E n /B; Z) W P 2 = w W P Zσ w Z[ 1 p ][σ [2], c 2,...,c n ] H (E n /P 2 ; Z[ 1 p ]), (p : torsion primes)

30 W H (G/B; Z) { σ w (l(ws i ) = l(w) + 1)) s i (σ w ) = σ w (α,α i )σ wsi s α (l(ws i ) = l(w) 1)) ( α l(ws i s α) = l(w) ) i {1,...,l}, i : H (G/B; Z) H 2 (G/B; Z) i (f g) ( = i (f) g + s i (f) i (g), f, g H (G/B; Z) σ wsi (l(ws i ) = l(w) 1) i σ w = 0 (l(ws i ) = l(w) + 1) w = [i 1,...,i k ] W w = i1 ik : H (G/B; Z) H 2l(w) (G/B; Z) f(σ vi ) = w (f)σ w

31 E n /P 2 Z[σ [2], c 2,..., c n] Im `Z[σ [2], c 2,..., c n] H (E n/p 2 ; Z) σ wk Z[σ [2], c 2,..., c n, σ wk ] ker( w) r k H (E n /P 2 ; Z) = Z[σ w k ] (r k )

32 E 6 /P 2 deg W P2 [2] [4, 2] [3, 4, 2],[5, 4, 2] [1, 3, 4, 2],[3, 5, 4, 2],[6, 5, 4, 2] G 2 := {σ [2], c 2, c 3, c 4, c 5, c 6 }: generator set R 2 := : relation set degree 4 1 σ 2 [2] = σ [4,2] 2 c 2 = 4σ [4,2] 3 G 4 := {σ [2], ĉ 2, c 3, c 4, c 5, c 6 } 4 R 4 := {g 2 = 4σ 2 [2]} degree 6 1 σ 3 [2] = σ [3,4,2] + σ [5,4,2] 2 c 3 = 2σ [3,4,2] + 4σ [5,4,2] 3 G 6 := {σ [2], σ [5,4,2], ĉ 3, c 4, c 5, c 6 } 4 R 6 := {g 2, g 3 = 2σ [5,4,2] + 2σ 3 [2] } H 6 (E 6 /P 2 ; Z) = Z[σ [2],σ [5,4,2] ]

33 GL n (C)/B ι E n /B p E n /P 2

34 GL n (C)/B ι E n /B p E n /P 2 Z[t 1,...,t n ] H (GL n (C)/B); Z)

35 GL n (C)/B ι E n /B p E n /P 2 Z[t 1,...,t n ] H (GL n (C)/B); Z) H (E n /P 2 ; Z) = Z[σ w k ] (r k )

36 GL n (C)/B ι E n /B p E n /P 2 Z[t 1,...,t n ] H (GL n (C)/B); Z) H (E n /P 2 ; Z) = Z[σ w k ] (r k ) c i H (E n /P 2 ; Z) g i H (E n /B; Z) = Z[t 1,...,t n ] H (E n /P 2 ; Z), (i = 1,...,n) c i g i (Nakagawa-K) H (E n /B; Z) = Z[σ [i],σ wk ], (i = 1,...,n) (c i g i, r k )

37 σ w w W w = u v, u W P, v W P l P (w) := l(v)

38 σ w w W w = u v, u W P, v W P l P (w) := l(v) { σ w = σ uv H σ u H (G/P) (G/B) σ v H (P/B) u W P v W P

39 σ w w W w = u v, u W P, v W P l P (w) := l(v) { σ w = σ uv H σ u H (G/P) (G/B) σ v H (P/B) u W P v W P u W P σ u H (G/P; Z):

40 σ w w W w = u v, u W P, v W P l P (w) := l(v) { σ w = σ uv H σ u H (G/P) (G/B) σ v H (P/B) u W P v W P u W P σ u H (G/P; Z): v W P σ v H (P/B; Z):

41 σ w w W w = u v, u W P, v W P l P (w) := l(v) { σ w = σ uv H σ u H (G/P) (G/B) σ v H (P/B) u W P v W P u W P σ u H (G/P; Z): v W P σ v H (P/B; Z): (Transition ) σ w = σ u σ v w :l P (w )<l P (w) w (σ u σ v )σ w l P (w) = 0 σ w H (G/P; Z)

42 H (G/B; Z) = Z[σ [i],σ wk ] (r k ) ( ) σ wk : G Chow A (G) A (G) (r k ): W W stable invariants H (G; Z) [Duan-Zhao] H (G; F p ) A p [Duan-Zhao]

43 torsion index t(w) N, t(w)σ w Im (H (BT; Z) H (G/B; Z)) decomposability σ w : σ w indecomposable σ w transition Giambelli T - orbifold Schubert calculus T H = L Z/λ i Z G/P G/P(λ) := H\G/P (weighted projective space ) H (G/P(λ); Q) = H (G/P; Q) H (G/P(λ); Z)

: : : : ) ) 1. d ij f i e i x i v j m a ij m f ij n x i =

: : : : ) ) 1. d ij f i e i x i v j m a ij m f ij n x i = 1 1980 1) 1 2 3 19721960 1965 2) 1999 1 69 1980 1972: 55 1999: 179 2041999: 210 211 1999: 211 3 2003 1987 92 97 3) 1960 1965 1970 1985 1990 1995 4) 1. d ij f i e i x i v j m a ij m f ij n x i = n d ij