S n Lie sl n (C) :={ n n } C (i) (ii) V V {} Specht Lie sl n (C) -p Hecke - Lie 98 -Drinfeld Lie - Hecke Lie () - v, q Hecke H n (q) C U v (sl n ) C L
|
|
- あきお もてぎ
- 5 years ago
- Views:
Transcription
1 Hecke Introduction V k V GL(V ) V End k (V ) k- Lie Gk- Ak Lie g G GL(V ), A End k (V ), g End k (V ) k-lie [] / -- Langlands / [] Fourier review gl n Hecke Lascoux-Leclerc-Thibon-Ariki Kazhdan-Lusztig enomoto@math.kyoto-u.ac.jp
2 S n Lie sl n (C) :={ n n } C (i) (ii) V V {} Specht Lie sl n (C) -p Hecke - Lie 98 -Drinfeld Lie - Hecke Lie () - v, q Hecke H n (q) C U v (sl n ) C Lie () sl n C n (C n ) m S m (C n ) m sl n S m
3 Hecke 3 Schur-Weyl U v (sl n ) Hecke H m (q) Hecke () Hecke C q Grothendieck affine Lie affine Lascoux-Leclerc-Thibon-LLTA Lie v Hecke Kazhdan-Lusztig review C Specht / affine Lie 4 S n 995 Kleshchev Lascoux-Leclerc-Thibon 5 Hecke Hecke q Hecke 6 U v (sl ) U v (ŝl l) Fock melting 7 A Hecke LLTA LLTA
4 4 LLT [LLT96] [Iw],[O],[TH] [FH],[Ful],[Mac],[Sa],[JK] Hecke [Mat] [A:book] Hecke [LLT96] [Kl:book],[DDPW] [BR] Lie review [TN] [ARS] [ASS] [HK] [K:book] [Kas9],[Kas93] LLT [A:book] [Ari96] graded representation theory review [Kle] [CG] [HTT] Lusztig [L:book] [Lus]
5 Hecke 5 Introduction Specht Robinson-Schensted S n Kleshchev Lascoux-Leclerc-Thibon Hecke Hecke A -Hecke (q-specht ) A -Hecke U v (sl ) U v (ŝl l) Fock F lower Lascoux-Leclerc-Thibon A Hecke LLTA
6 6. k G G.. (i) V k G V ρ : G GL k (V ) dim k V (ii) G V G-V W ρ(g)w W g G ρ : G GL k (W ) V G ρ V {} G- (iii) G (ρ,v ), (ρ,v ) ρ : G GL k (V V ); g ρ (g) ρ (g) V V (iv) G (ρ,v ), (ρ,v ) φ : V V φ ρ (g) = ρ (g) φ g G V φ V ρ (g) V φ V ρ (g) V G G C. (). S n ρ : S n GL C () = C s i =(i, i + ) ( i n ) s i =( i n ), s i s i+ s i = s i+ s i s i+, s i s j = s j s i ( i j > ). ρ =ρ(s i ) ρ(s i ) {±} = ρ((s i s i+ ) 3 ) =ρ(s i )(s i+ ) S n i ρ(s i )= σ S n ρ(σ) = i ρ(s i )= σ S n ρ(σ) = sgn(σ).3 (). S n C n s i i, i + ρ : S n GL C (n); s i i n S n I n i, (I r r ) n
7 Hecke 7 W := {(a, a,..., a); a C}, W := {(x,x,...,x n ); x + x + + x n =}. W W C n = W W.4 (). G X e x k- kx := ke x G x X ρ : G GL k (kx); ρ(g)e x := e g x (g G, x X). X = G G () G = S = {id, ()} CS = Ce id Ce () S ρ ρ(id)(e id ± e () )=e id ± e (), ρ(())(e id ± e () )=e () ± e id CS = C(eid + e () ) C(e id e () )..5. G (ρ, V ) (ρ, V ).6 (Maschke ). G k G G k A A G
8 8 3 (i) G (ρ,v ), (ρ,v ) ρ ρ : G GL k (V V ) ρ ρ (g)(v v ):=ρ (g)v ρ (g)v A Lie Hecke (ii) G H G (ρ, V ) H ρ : H GL k (V ) G ρ Res G H V Res G H ρ H (ρ,w) kg kh W ( = k(g/h) Was) 4 G ρ Ind G H W Ind G H ρ Frobenius Hom H (Res G H V,W) = Hom G (V,Ind G H W ) Z- Grothendieck A A A A n A i Grothendieck K n K := K n n Grothendieck 4 / /
9 Hecke 9.3 G character.7. G (ρ, V ) χ V : G C χ V (g) :=tr(ρ(g)) (ρ, V ) (ρ, V ) χ V (id) = dim V χ V G G G V,V χ V V = χ v + χ V χ V V = χ V χ V G ϕ, ψ : G C ϕ, ψ := ϕ(g)ψ(g). G g G.8. G (i) G V χ V,χ V = (ii) G V W χ V,χ W (iii) G G.9 (S 3 ). 3 S 3 6 id, (), (3), (3), (3), (3) s = (),s (3) id,s,s,s s,s s,s s s (= s s s ) {id}, {s,s,s s s }, {s s,s s } 3 ρ(σ) S 3 S 3 GL C (3) ρ(id) 3 s = () s s = (3) {id} {s,s,s s s } {s s,s s } χ 3 χ triv χ sgn χ χ triv,, χ, χ triv = ( ) = 6 χ χ triv,χ χ triv = ( ) = 6
10 χ χ triv χ χ triv,χ sgn = ( ) = 6 S 3 3 (character table) S 3 {id} {s,s,s s s } {s s,s s } χ triv χ sgn. (). G ρ : G GL C (CG) ρ(g)e g = e gg g e gg e g χ ρ (id) = G id g G χ ρ (g) = G W χ ρ,χ W = G χ ρ(id) χ W (id) = χ W (id) = dim W CG. (). Irr G G CG = G = V Irr G (dim V ) V i Irr G V dim V i i. (S 4 ). 4 S 4 5 id, (), ()(34), (3), (34) ρ. 4 = S 4 = (dim ρ) dim ρ = Specht = ρ, χ triv = ( + 6a +3b +8c +6d) 4 = ρ, χ sgn = ( 6a +3b +8c 6d) 4 = ρ, χ standard = (6 + 6a 3b 6d) 4 = ρ, ρ = 4 (4 + 6a +3b +8c +6d ) a, b, c, d S 4 id () ()(34) (3) (34) 3 ρ 3
11 Hecke.3. 3 (i) S 3 S 3 S 3 id () (3) standard standard 4 standard standard = triv standard sgn (ii) S 4 4 S 3 S 4 S 3 S 4 id, (), (3) S 4 S 3 S 3 id () (3) Res S 4 S 3 (standard) 3 S 3 Res S4 S 3 (standard) = triv standard 3 n S n σ λ,λ,... λ =(λ λ λ 3 ) λ i Z λ i = n n n (partition) λ n 3.. S n n G G S n n n λ 3. (S n ). i Specht Young Springer CS n
12 3 4 4 n n n Jordan (up to Jordan ) sl n (C) sl n (C) Specht Irr S n : S n : {n } Springer : {sl n (C) } Hecke 3. Specht n Young n λ =(λ λ ) λ box λ box box Young λ =(7, 5, 5, 4, ) Specht Young box 3.3. λ n (i) λ (tableau) λ Young box n (ii) λ (standard tableau) box λ n n λ shape : 3, 3, : 3, 3, 3, 3 Specht 3.4. λ n λ B k box {b k,b k,,b kr } f B (X,...,X n )= k Specht (X bki X bkj ) i<j r
13 Hecke 3 B = , f B (X,...,X 6 )=(X 3 X 5 )(X 3 X )(X 5 X ) (X 4 X 6 ) n C[X,X,...,X n ] n S n (σ f)(x,x,...,x n )=f(x σ (),X σ (),...,X σ (n)) C[X,X,...,X n ] S n Specht f B σ B σ Specht S n C[X,X,...,X n ] 3.5. S λ := C-span f B (X,...,X n ) B λ C[X,X,...,X n ] S n Specht f B B (i)s λ {f B B λ } C-dim S λ = {λ } (ii) S λ S n (iii) S λ Z Z 3.7. n (n) ( n ) Young (5) : 3 4 5, ( 5 ): Specht (X i X j ) S (n) S (n) 3.8. n =3,λ=(, ) i<j B : B : B 3 : 3 3 3, f B = X X, s f B = X X = f B, s f B = X X 3 = f B,, f B = X X 3, s f B = X X 3 = f B + f B, s f B = X X = f B,, f B3 = X X 3 = f B + f B ( s = ), s = ( ) (, s s = ).
14 4 χ χ (id) =,χ () =,χ (3) = S n =4,λ=(, ) B : B : , f B =(X X 3 )(X X 4 ), f B =(X X )(X 3 X 4 ) ()f B =(X X 3 )(X X 4 )=f B f B ( s = ) (, s = ) (, s 3 = ). χ S 4 id () ()(34) (3) (34) χ. S Specht Specht S λ S 3 S 3.8 s, Specht f B,f b f B s f B Specht S λ Res S n S n = S µ S λ {f B } S µ 3.. S λ λ hook λ λ box x (hook) x hook h x =5 x
15 3. (). λ Hecke 5 dim S λ n! = x λ h x (5, 5, 3, ) 4 box hook dim S (5,5,3,) = 4! = Robinson-Schensted CS n.(iv) CS n = λ n(s λ ) dim Sλ dim S λ = {λ } n! = ( {λ }) λ n 3.3 (Robinson-Schensted ). n {,,...,n} shape n ( ) σ = k k box k k shape (4,, ) (4,, ) box step σ (P, Q) bumping procedure =: Q =: P 3.3 S n Specht
16 6 C[X,...,X n ] S n C[X,...,X n ] Sn d C[X,...,X n ] S n d ρ d m,n : C[X,...,X m ] S m d Sym d := lim C[X,...,X n ] S n d { C[X,...,X n ] S n d ; X Xi (i n) i (i>n) Sym = d Sym d α =(α,...,α n ) N n X α = X α X α n n a α (X,...,X n )= sgn(w)w(x α ) w S n 3.5. λ n δ =(n,n,...,, ) λ Schur s λ (X,...,X n )= a λ+δ a δ 3.6. λ s λ λ = i λ i ρ n+,n s λ (X,...,X n,x n+ )= s λ (X,...,X n ) s λ Sym well-defined {s λ λ } Sym {s λ λ d} Sym d Schur Schur 3.7. λ n λ n (semi-standard tableau) λ B a i = B i wt B =(µ,µ,,µ n ) 3.8. λ n s λ (X,X,,X n )= B:λ X wt B B λ X wt B := X µ Xµ Xµ n n =3 3 (3,, ), (,, ), (,, ) Schur (i) (3,, ),, 3,, 3, 3 3,, 3, 3 3, 3 3 3
17 Hecke 7 s (3,,) (X,X,X 3 )=X 3 + X X + X X 3 + X X + X X X 3 + X X 3 + X 3 + X X 3 + X X X (ii) (,, ), 3,, 3, 3, 3 3, 3, s (,,) (X,X,X 3 )=X X + X X 3 + X X +X X X 3 + X X 3 + X X 3 + X X 3 (iii) (,, ) 3 s (,,) (X,X,X 3 )=X X X r p r := i Xr i Sym λ =(λ λ ) p λ = p λ p λ 3.. {p λ λ } Sym S n Z- K(S n ) S λ [S λ ] K(S n ) K = n K(S n) λ n, µ m [S λ ] [S µ ]= [ Ind Sn+m S n S m S λ S µ] K = n K(S n) K Z C Sym 3. (e.g.[mac]). ch : K Z C Sym; [S λ ] s λ p ρ = λ χλ ρs λ χ λ ρ ρ S λ χ λ 3.3. K Sym [Mac] 3.4. Schur 3 p = X + X + X 3, p = X + X + X 3 3, p 4 = X 3 + X 3 + X 3 3 p (3,,) = X 3 + X 3 + X 3 3 = s (3,,) s (,,) + s (,,), p (,,) =(X + X + X 3 )(X + X + X 3 3 ) = X 3 + X X + X X 3 + X X + X X 3 + X 3 + X X 3 + X X X 3 3 = s (3,,) s (,,), p (,,) =(X + X + X 3 ) 3 = X 3 + X 3 + X (X X + X X + X X 3 + X X 3 + X X 3 + X X 3 )+6X X X 3 = s (3,,) +s (,,) + s (,,)
18 8 χ λ ρ.9 S 3 ( 3 ) (,, ) (3) S 3 (), (3), (3) (3), (3) s (3) s (,) s (,,) 3.4.3(ii) Specht n Specht S λ S 3 id (), (3), (3) (3), (3) S S S S 4 id () ()(34) (3) (34) S S 3 S S 3 S Res S4 S 3 S λ Young.3(ii) id, (), (3) S 4 \S 3 id () (3) S S S S S 3 S S 3 3 S Res S 4 S 3 S = S, Res S 4 S 3 S = S S Frobenius Ind S4 S 3 S = S S S Young
19 Hecke (S n ). S n n S n (i) λ n Res S n S n S λ = S µ µ Young λ box µ Young (ii) ν n Ind Sn S n S ν = S µ µ Young ν box Young µ 3.7. S λ S µ Specht 3. Young 3.8. λ Young box Young box removable box Young box addable removable addable 3.9. µ addable box λ µ λ Young Young 3.3. box x content (x )-(x ) l Z x content (mod l) x l-residue l-residue i addable,removable box i-addable,i-removable box content -residue 3-residue 4-residue Young l-residue= i box i ( i l ) µ i λ
20 φ Young n ( Young ). l = φ
21 Hecke l =3 φ l Z Grothendieck K Z C = n K(S n) Z C e i f i e i [S λ ]= [S µ ], f i [S µ ]= [S λ ] ( i l ) µ λ i µ λ i h i [S λ ]=( {i-addable box to λ} {i-removable box in λ})[s λ ] {e i,h i,f i ; i l } l =,λ=(3, ) [S λ ] λ f + e + + e f + + (e f f e )[S (3,) ]=[S (3,) ]=h [S (3,) ] (h e e h )[S (3,) ] = [S (,) ]=e [S (3,) ], (h f f h )[S (3,) ]= ([S (3,) ]+[S (3,,) ]) = f [S (3,) ], e,h,f [f,g] =fg gf Lie [e,f ]=h, [h,e ]=e, [h,f ]= f
22 Lie sl (C) e,h,f sl K Z C e f e h e e h f f h e e [e,f ]= [h,e ]= e l =3 h,e e + h + h e + [h,e ]= e A () l affine Lie Kac-Moody Lie Lie A () l Cartan l l A = ( ). (l = ), A = (l 3) A =(a ij ) i,j l e i,h i,f i ( i l ) Lie ŝl l A () l affine Lie [h j,e i ]=a ij e i, [h j,f i ]= a ij f i ( i, j l ) [h i,h j ]=, [e i,f j ]=δ ij h i ( i, j l), [e i, [e i,e j ]] = (j i ± mod l), [e i,e j ] = (otherwise), [f i, [f i,f j ]] = (j i ± mod l), [f i,f j ] = (otherwise) ( ). {e i,h i,f i i l } affine Lie ŝl l K Z C Fock e i,f i K( Z C) virtual Jucys-Murphy
23 Hecke CS n S n L k := (,k)+(,k)+ +(k,k) CS n ( k n) Jucys-Murphy L k S k CS n L n S λ S λ Specht 3.4. (i) λ =(, ) L 3 = (3) + (3) L 3 f 3 L 3 f 3 = L 3 (X X )=(X 3 X )+(X X 3 )=X X = f 3,. = L 3 (X X 3 )=(X 3 X )+(X X )= f 3 + f 3 (ii) λ =(, ) L 4 = (4) + (4) + (34) L 4 f 3 4 L 4 f 3 4 = L 4 (X X 3 )(X X 4 ) =(X 4 X 3 )(X X )+(X X 3 )(X 4 X )+(X X 4 )(X X 3 ) ( ) = f 3 4 f f 3 4 f 3 4 = = L 4 (X X )(X 3 X 4 ) =(X 4 X )(X 3 X )+(X X 4 )(X 3 X )+(X X )(X 4 X 3 ) ( ) = f f f 3 4 f 3 4 = (iii) λ =(,, ) L 4 = (4) + (4) + (34) L 4 f 4 3 L 4 f 3 4 L 4 f 3 4 = f 4, 3 = f 3 4 = f f 4, 3. f 4 3 (iv) λ =(3, ) L 5 = (5) + (5) + (35) + (45) L 5 f = f f L n f B B B n B f B B n content 3.4. λ n λ removable box (i,j ), (i,j ),...,(i r,j r ) µ (k) := λ\(i k,j k ) (n ) S λ L n (j i, dim S µ() ), (j i, dim S µ() ),...(j r i r, dim S µ(r) ),
24 4 L n S n S n S µ(k) ( k r) i l e i : S n -mod S n -mod; V k i (m o d l) (V L n k ) K e i f i L n Res Sn S n S λ 3.7 Jucys-Murphy Hecke LLTA A K Z C Sym affine Lie ŝl l Sym [S λ ] Schur s λ Schur ŝl l Lie n K(S n ) Fock affine Lie ŝl l K Z C = n K(S n) Z C Sym affine Lie Virasoro affine Lie Virasoro Schur Jack
25 Hecke Specht Z S λ Z F p F p F 4.. F S n C F F 4.. F S F S F S = {,e id,e (),e id + e () } 4 U := {,e id + e () } F U F = U W S W F () e id = e (), () e () = e id {,e id }{,e () } U U W F S /U F S (indecomposable) 4.3. S 4 C (3, ) 3 4, 4 3, Specht C S (3,) {X X, X X 3, X X 4 } 3 S 4 s,s,s 3 s =, s =, s 3 = S (3,) Z F F 8 Specht {, X + X,X + X 3, X + X 4, X + X 3, X + X 4, X 3 + X 4,X + X + X 3 + X 4 }
26 6 8 S 4 W := {,X + X + X 3 + X 4 } s =, s =, s 3 = 3 s,s,s 3 S 4 Z C S λ F p S λ Z F p W X i +X j 8 X +X s,s 3 X +X 3,X +X 4 X +X 3,X +X 4,X 3 +X 4,X +X +X 3 +X 4 S (3,) Z F /W 4.4. S 3 S (,) C {X X,X X 3 } S (,) Z F 4 {,X + x,x + X 3,X + X 3 } {,X i + X j } S 4 S (3,) F 3 4. k G C affine Hecke Lie Hecke A..
27 Hecke 7 3 Schur V End A (V ) V End A (V ) Krull-Schmidt ADE -tame tame wild A A A A * Hecke Lie Verma =hook 4.5. A V (composition series) V = V V V N = {} i N V i /V i+ {V i /V i+ ( i N )} (composition factor) (composition multiplicity) V 4.6 (Jordan-Hölder). * standard
28 (i) Lascoux-Leclerc-Thibon- Kazhdan-Lusztig (ii) A B A V Res A B(V ) soc Res A B(V ) soc Res A B(V ) Hecke Grothendieck Z- K(A) Grothendieck A V V = V V V N = {} N K(A) [V ]= [V i /V i+ ] i= Grothendieck K(A) A A A- A = B B r A- B i B,...,B r A A- Grothendieck - Broué Chuang-Rouquier sl -categorification 4.3 Kleshchev 4.3 C Specht S λ F p F p S n - Kleshchev 4.3 S (3,) Z F
29 Hecke λ Young p p-regular, -regular,, -regular 4.8 (James 975). S λ Z F p M Dp λ := S λ Z F p /M {Dp λ λ p-regular } Irr Fp S n 4.9. S (n) Z F p S (n) Z F p D p (n) ( = D n ) p {Dp λ λ n} λ p-regular 4.. F p Dp λ 4.. dim(dp λ ) S λ Z F p Res Sn S n D λ p Kleshchev Res Sn S n D λ p soc Res Sn S n D λ p 4.. λ i-addable box Ai-removable box R A, R AR R RA A R i-removable box i-good box λ i-good box µ µ i λ p-regular Kleshchev s p-good lattice 4.3. p =λ =(4, ) -residue i = -addable box -removable box A, R RR -good box R box -regular -good lattice p-good lattice p-regular box 4.4 (Kleshchev 995). soc Res Sn S n Dp λ Dµ p µ λ p-regular F p S n p-good lattice
30 p = p =3 p-good lattice p = good lattice φ
31 p = Hecke 3 φ
32 3 p =3 good lattice φ
33 p =3 Hecke 33 φ 4.6 (λ =(4, ) ). p =, 3 D (4,) S (4,) Z F p p =, 3 p =D (4,) Specht X + X,X + X 3,X + X 4,X + X 5 S 4 X +X,X +X 3,X +X 4 {,X +X +X 3 +X 4 } soc Res S 5 S 4 D (4,) = D (4) p =3D (4,) 3 Specht X +X,X +X 3,X +X 4,X +X 5 S 4 X +X,X +X 3,X +X 4 D (3,) 3 {,X +X +X 3 +X 4 +X 5 } Res S 5 S 4 D (3,) 3 D (4) 3 socle soc Res S 5 S 4 D (4,) 3 = D (4) 3 D (3,) 3 p =,p=3 4.4 Lascoux-Leclerc-Thibon Kleshchev p-good lattice Lascoux-Leclerc-Thibon
34 34 98 Drinfeld U v (g) Kac-Moody Lie g U(g) - Hopf - v U(g) affine Lie ŝl l Fock affine U v (ŝl l) F := C(v) λ n λ n Young L(Λ ):=U v (ŝl l) φ (crystal base) U v (g) v U v (ŝl l) L(Λ ) Misra-Misra- Lascoux-Leclerc-Thibon Kleshchev p-good lattice Lascoux-Leclerc-Thibon (995) Kleshchev p-good lattice affine U v (ŝl l) L(Λ ) (global base) Lusztig (canonical base) v bar involution L(Λ ) Lascoux-Leclerc- Thibon affine U v (ŝl l) L(Λ ) Hecke Hecke
35 Hecke 35 5 Hecke 5. Hecke 5.. q C A Hecke H n (q) C- : T,T,...,T n :(T i q)(t i + ) = T i T i+ T i = T i+ T i T i+ ( i n ) T i T j = T j T i ( i j > ). 5. (H n (q) ). w S n reduced expession w = s i s ir T w = T i T ir w reduced expression {T w ; w S n } H n (q) C Hecke 96 s p GL n (Q p ) Lie GL n (F q ) Bruhat Weyl Coxeter q-analogue U v (g) Lie U(g) Hopf v--hecke CW Hopf q- H n (q)- V,W =H n (q) q- Weyl Hecke (i) H =End G (Ind G B()) (ii) gl n Schur-Weyl (iii) Jones KZ (iv) Kazhdan-Lusztig Kazhdan-Lusztig Lie (v) affine double affine Macdonald (vi) K- (Kazhdan-Lusztig, Ginzburg) Hodge (vii) (Khovanov-Lauda,Brundan-Kleshchev) (viii) q-schur rational Cherednik 5.4 (Kazhdan-Lusztig ). Kazhdan-Lusztig A Coxeter Hecke {T w } w W {T w } w W Kazhdan-Lusztig Kazhdan-Lusztig 5.5. (W, S) Coxeter l(w) w W S Bruhat W Kazhdan-Lusztig P y,w (q) W y, w
36 36 (i) W Bruhat y w P y,w (q) =y = w P y,w (q) =y <w l(w) l(y) q (ii) W Hecke C w := q l(w)/ P y,w (q)t y y w Hecke q q,t w T w {C w } w W Hecke Z[q /,q / ]- Kazhdan-Lusztig q = P y,w () W G Borel X/B Bruhat w W stratification X/B = w W X w X w Schubert X w Schubert Kazhdan-Lusztig P y,w (q) = ( ) q i dim IHX i y (X w ) i Kazhdan-Lusztig Lie Kazhdan- Lusztig W w ρ W W W M w w(ρ) ρ Verma w(ρ) ρ L w M w M w L w Lie ch 5.6 (Kazhdan-Lusztig ). ch(l w )= ( ) l(w) l(y) P y,w () ch(m y ), ch(m w )= P w w,w y() ch(l y ). y w y w Beilinson-Bernstein,Brylinski- D- Riemann-Hilbert Verma ρ Lie Bernstein-Gelfand-Gelfand Verma Kazhdan-Lusztig q = 5. A -Hecke (q-specht ) 5.7. H n (q) C[X,...,X n ] T i (X m X m n n )= qx m X m n n (m i = m i+ ) X mi+ i X mi i+ Xm n n +(q )X m X mi i X mi+ i+ X m n n (m i <m i+ ) X m i+ i X m i i+ Xmn n (m i >m i+ ) qx m X m
37 Hecke 37 λ n,, 3,... B λ =(3, 3, ), B = B q- f B (X,...,X n ):= k (qx i X j ) i<j B k λ B w B = B w S n f B (X,...,X n ):=T w f B (X,...,X n ) q-specht f B (X,...,X 7 )=(qx X )(qx X 3 )(qx X 3 )(qx 4 X 5 ) (qx 6 X 7 ) 5.9. S λ q := C-span f B ; B λ C[X,...,X n ] H n (q) (i) q S λ q {f B; B λ } C- {S λ q λ n} H n (q)- (ii) q l-s λ q Dλ q {D λ q λ l-regular n } H n (q) 5.. Hecke q l p 5.. A -Hecke q-specht q- q-specht q- Young q- cellular CS n
38 A -Hecke 5.9 l = p F p S n H n ( p ) p-regular n * 5.. soc Res H n ( l ) H n( l D λ l ) Dµ l µ λ l-regular l-regular l-good lattice F p S n H n ( p ) Hecke l S λ S λ Z F p [S λ Z F p : D µ p ] Hecke H n (q) S λ q q l [S λ l : Dµ l ] l = p S λ Z F p S λ l 5.3. n =5,λ=(3, ) λ 3 5 4, 3 4 5, 5 3 4, 4 3 5, Specht F S (3,) f =(X + X )(X 3 + X 4 ), f =(X + X )(X 3 + X 5 ), f =(X + X 3 )(X + X 4 ), f 3 =(X + X 3 )(X + X 5 ), f 4 =(X + X 4 )(X + X 5 ) f +f +f 3 +f 4 = X X +X X 3 +X X 4 +X X 5 +X X 3 +X X 4 +X X 5 +X 3 X 4 +X 3 X 5 +X 4 X 5 = s,s,s 3,s 4 s i s =,s = s 3 =,s 4 =, i<j 5 X i X j * LLTA
39 S (3,) Z F 5.4. n =5,λ=(3, ) λ Hecke 39 B = s 4 B = s B = s 4 B 3 = s 3 B 4 = q-specht f B =(qx X )(qx 3 X 4 )=q X X 3 qx X 3 qx X 4 + X X 4 f B = q 3 X X 3 q X X 3 qx X 5 + X X 5 f B = q 3 X X + q (q )X X 3 q X X 3 q X X 4 + X 3 X 4 f B3 = q 4 X X + q 3 (q )X X 3 q 3 X X 3 q X X 5 + X 3 X 5 f B4 = q 5 X X + q 3 (q )X X 4 q 3 X X 4 q 3 X X 5 + X 4 X 5 T,T,T 3,T 4 q q 4 q q 3 T = q q,t = q q q q T 3 = q q,t 4 = q q q 4 q q q 3 q q q q 4 q q 3 q q q q = T i v = v v S (3,), Hecke Grothendieck n 5.5. D(F p S n )=(d λµ ), d λµ =[S λ Z F p : D µ p ] (λ, µ n, µ p-regular) (decomposition matrix) Hecke D(H n ( l )) = (d λµ ), d λµ =[S λ : l Dµ l ] (λ, µ n, µ l-regular) A -Hecke
40 4 5.6 (n =5) S 5 (F ) D (5) D (4,) D (3,) S (5) 4 S (4,) 5 S (3,) 6 S (3,,) 5 S (,,) 4 S (,,,) S (,,,,) 4 5 H 5 ( ) D (5) D (4,) D (3,) S (5) 4 S (4,) 5 S (3,) 6 S (3,,) 5 S (,,) 4 S (,,,) S (,,,,) F S 5 - S λ Z F H 5 ( )- S λ F S 5 - H 5 ( )-
41 Hecke 4 6 U v (ŝl l) Fock 6. U v (sl ) 6.. v U v (sl ) E,F,K ± C(v)- KK ==K K, KEK = v E, KFK = v F, EF FE = K K v v. K = v H v [H, E] =E, [H, F] = F, [E,F]=H Lie sl U v (sl ) Lie sl 6.. Lie g V,W V W g End(V W ); X (v X(v) w + v X(w)) Lie g U(g) X X + X U(g) Hopf g V W v w w v W V g U(g) Hopf U v (sl ) V,W V W K(v w) =Kv Kw, E(v w) =Ev K w + v Ew, F (v w) =Fv w + Kv Fw U v (sl ) (K) =K K, (E) =E K + E, (F )=F +K F U v (sl ) Hopf V W v w w v W V U v (sl ) U v (sl ) Hopf R : V W W V U v (sl ) R R- Drinfeld- R Kac-Moody Lie g U v (g) affine Lie ŝl l U v (sl ) v
42 U v (sl ) V E,F N E N V =,F N V =. K V Ev = cv v V,c EK = v K E E(K j v)=v j K j Ev = cv j K j v K j v (j Z) E {cv j } j Z E v c E V E F V Ker E Ker E u Ker E EKu = v KEu = Ku Ker EM K Ku = cu, Eu = u V,c C(v) F m u F m+ u = m Z EF m+ = F m+ E + EF m+ u = F m (v v ) {(v v m )K +(v v m+ )K } (v v m )c +(v v m+ )c (v v ) = c = ±v m [j] v := vj v j v v { F v, Fu, [] v! u, F 3 [3] v! u,..., F m } [m] v! u K (±v m, ±v m,...,±v m ) V U v (sl ) m + V (m) ± E,F,K [m] v [m ] v E = ± K = ± v m v m v m+... [] v v m,f = [] v [] v [m] v W K W F j [j] v u W E j F j [j] v u =[m j + ] v [m j + ] v [m] v u u W u V W = V,
43 Hecke v {V (m) ± } m Z U v (sl ) 6.5. v U v (sl ) 6.6. Casimir C := FE vk v K (v v ) E,K,F U v (sl ) V (m) ± ± [m + ] v v v 6.7. U v (sl ) ι ι(e) = E, ι(f )=F, ι(k) = K U v (sl ) V (m) ι V (m) + U v (sl ) V () V (m) =V () V (m) + v U v (sl ) U v (g) K v 6. U v (ŝl l) 6.8. v U v (ŝl l) E i,f i,k ± i C(v)- K i K j = K j K i, K i K i ==K i K i, ( i l ) K i E j K i = v a ij E j, K i F j K i = v a K ij i K i F j, [E i,f j ]=δ ij v v, l 3 E i E j (v + v )E i E j E i + E j Ei = (j i ± mod l), E i E j = E j E i (otherwise), F i F j (v + v )F i F j F i + F j F i = (j i ± mod l), F i F j = F j F i (otherwise), l =E 3 i E j (v ++v )E i E j E i +(v ++v )E i E j E i E j E 3 i =(i j), F 3 i F j (v ++v )F i F j F i +(v ++v )F i F j F i F j F 3 i =(i j). i {E i,f i,k ± i } U v (sl ) i 6.3 Fock P P n n λ C(v)- F := C(v) λ = C(v) λ λ P n λ P n
44 44 Fock λ = µ {x} N i N i (λ\µ) = {x λ i-addable box} {x λ i-removable box} (λ\µ) = {x λ i-addable box} {x λ i-removable box} N i (λ) = {λ i-addable box} {λ i-removable box} x l-residue res l 6.9 (). U v (ŝl l) E i,f i,k i ( i l ) E i λ = v N i (λ\ν) ν, F i λ = v N i (µ\λ) µ, res l (λ\ν)=i res l (µ\λ)=i K i λ = v N i(λ) λ F U v (ŝl l) F U v (ŝl l)- L(Λ ):=U v (ŝl l) φ F 6.. l = E φ =, E φ =, F φ =, F φ = (), E () =, E () = φ, F () = () + v (, ), F () =, E () = v (), E () =, F () = v (, ), F () = (3), E (, ) = (), E (, ) = F (, ) = (, ) F (, ) = (,, ) E (3) = E (3) = () F (3) = (4) + v (3, ) F (3) = E (, ) = () + v (, ) E (, ) = F (, ) = F (, ) = (3, ) + v (, ) + v (,, ) E (,, ) = E (,, ) = (, ) F (,, ) = (,, ) + v (,,, ) F (,, ) = 6.4 F Fock U v (ŝl l) V () {K,K,...,K l } v () E i,f i u V N Ei N u =,Fi N u = () () u V U v (sl ) i - V U v (sl ) i
45 F E i,f i,k ± i = U v (sl ) k + V (k) + = k m= Hecke 45 C(v)u (k) m ( E i u (k) = F i u (k) m =, u m = F i m ) [m] v! u(k) v- [m] v := vm v m v v F i u (k) m = u (k) m+, Ẽ i u (k) m = u (k) m F 6.. A v = C(v) L := λ P A λ, B F := { λ mod vl} L/vL (i) ẼiL L F i L L (ii) B F L/vL (iii) Ẽi : B F B F {} F i : B F B F {} (L, B F, Ẽi, F i ) F b B F Ẽib F i Ẽ i b = b F i b Ẽi F i b = b F i b = b b F i b B F 6.. l = E,F,K ± E V k V k u (k) m E φ =, φ () E () =, () (3) E (, ) =, (, ) (,, ) E (, ) =, (, ) (3, ) + v (, ) + v (,, ) (3, ) + v (3,, ) + v (,, ) (3,, ) E (4) =, (4) (5) E (,,, ) =, (,,, ) (,,,, ) E ( (, ) v (3, ) ) =, (, ) v (3, ) (,, ) v (3,, ) E ( (,, ) v (, ) ) =, (,, ) v (, ) (3,, ) v (3, )
46 46 E,F,K ± E V k V k u (k) m E φ =, φ E () =, () () + v (, ) (, ) E ( (, ) v () ) =, (, ) v () E (3) =, (3) (4) + v (3, ) (4, ) E (,, ) =, (,, ) (,, ) + v (,,, ) (,,, ) E (, ) =, (, ) E ( (,,, ) v (,, ) ) =, (,,, ) v (,, ) E ( (3, ) v (4) ) =, (3, ) v (4) L(Λ ) φ π : F L(Λ ) L/vL L /vl L = L L(Λ ) B = πb F B = {π( λ (mod vl)) λ l-regular n } (L, B ) (i)-(iii) L(Λ ) B B φ (mod vl) -regular Kleshchev -good lattice B F B Misra- 6.3 (Misra-). B F = { λ mod ql} Ẽi, F i λ i-good box Ẽi λ = modql x λ i-good box µ := λ\{x} Ẽ i λ = µ mod ql, Fi µ = λ mod ql µ {x} x i-good box x F i µ = modql
47 Hecke 47 Kleshchev s p-good lattice Lascoux-Leclerc-Thibon 6.4. l = p L(Λ ) Kleshchev p-good lattice 6.5. U v (ŝl l) E i,f i,k ± i U v (g) g U v (sl ) U v (g) M M B i U v (sl ) i - M 6. E,F,K ± ( φ () ) ( () (3) ) ( (, ) (,, ) ) E,F,K ± U v(sl ) (, ) v () E,F,K ± Lie No v = (L, B, Ẽi, F i ) i Ẽi, F i : B B {} {B F := { λ mod vl} i U v (sl )- v = v = 6.5 lower U v (ŝl l) v = v, K i = K i, e i = e i, f i = f i ( i l ) bar involution u φ = u φ (u U v (ŝl l)) L(Λ ) bar involution [k] v := vk v k (k) v v f i := f i k [k]! U Q L(Λ ) Q := U Q f (k) i φ U v (ŝl l) Q[v, v ] L(Λ ) Q Q[v, v ]- {G low (µ) µ l-regular } (i) G low (µ) µ (ii) G low (µ) =G low (µ) (mod vl ) L(Λ ) lower
48 48 G low (µ) L(Λ ) F G low (µ) = λ d λµ (v) λ d λµ (v) Q[v, v ] 6.7. v melting bar 5.4 Kazhdan-Lusztig Hecke 6.8 (lower ). l = φ = φ f i = f i f φ =, f = + v, f ( + v ) = + v f ( + v ) = + v + v + v bar involution mod vl v µ (µ : -regular) G low (φ) = φ, G low ( )= G low ( )= + v G low ( )= + v G low ( )= + v + v + v f () ( = f ( ) = + v + v, f () ( ) = + v + v, f () ( + v ) = + v bar involution mod vl v µ (µ : -regular) G low ( )= G low ( )= + v + v, G low ( )= + v + v G low ( )= + v f ( + v + v + v ) = + +v + v + v
49 Hecke 49 bar involution mod vl v + (i) G low ( )= + v + v bar involution + v + v bar involution mod vl v µ (µ : -regular) G low ( ) = + v + v n =5 G low (µ) = λ d λµ (v) λ d λµ (v) d λµ G low ( ) G low ( ) G low ( ) (5) (4, ) (3, ) (3,, ) v v (,, ) v (,,, ) v (,,,, ) v v =H 5 ( ) [S λ : D µ ]
50 5 7 Lascoux-Leclerc-Thibon- 7. A Hecke lower G low (µ) = λ d λµ (v) λ A Hecke q = l [S λ : D µ ] Lascoux-Leclerc-Thibon ( 996). d λµ (v) N[v] d λµ () = [S λ l : Dµ l ] Kazhdan-Lusztig Ginzburg affine Hecke K- Lusztig 996 K v K L(Λ ) v = 7. Grojnowski-Vazirani K n ( Irr H n ( l ) ) Hecke Jucy-Murphy E i,f i K Grojnowski-Vazirani socle,cosocle Ẽi, F i L(Λ ) upper uuper {G up (λ) λ l-regular} lower L(Λ ) U v (ŝl l) E i G up (λ) = µ E i,λµ (v)g up (µ), F i G up (λ) = µ F i,λµ (v)g up (µ) v = E i,λµ (),F i,λµ () K- 7. (upper ). [ ) ] Res Hn H n (D λ l : D µ l = l E i,λµ (), i= [ ) ] Ind Hn H n (D λ l : D µ l = l F i,λµ (). i= [Dl λ ] upper v =K L(Λ ) v Hecke λ, µ upper λ, µ Kazhdan-Lusztig
51 Hecke 5 Grojnowski-Vazirani Grojnowski- Vazirani Ẽi, F i D λ l {λ λ l-regular Misra- 7.3 (). n Ẽ i D λ l = DẼiλ l, F i D λ l = D F i λ l. ( Irr H n ( l ) ) upper ε L(λ ) E i G up (λ) =G up (Ẽiλ) + higher term, F i G up (λ) =G up ( F i λ) + higher term Hecke upper categorification 8 Khovanov-LaudaRouquier graded reresentation theory Brundan-Kleshchev A Hecke v categorification v = K L(Λ ) U v (ŝl l) 7.4. graded representation theory 7. LLTA Lascoux-Leclerc-Thibon- A A A A n A n A n -mod Grothendieck K = K (A n -mod) n LLTA K 3 I. affine Lie categorification affine Lie K categorification A n -mod exact endfunctor K affine Lie n
52 5 II. Irr A n n III. K v = 7.3 LLTA A Hecke A Hecke [S λ Z F p : D p µ ] I Chuang-Rouquier n F ps n -mod sl categorification sl Weyl categorification tilting block Broué B Hecke cyclotomic Hecke A Hecke Hecke cyclotomic Hecke 996 G(m,,n) Hecke B Hecke L(Λ ) higher level graded representation theory establish affine Hecke A affine Hecke A Hecke cyclotomic Hecke C A affine Hecke LLTA affine Lie L(Λ ) Uv (ŝl l) LLTA establish A affine Hecke LLTA Miemietz 6 LLTA Varagnolo-Vasserot graded representation 9
53 Hecke 53 v-schur cyclotomic v-schur Cherednik v-schur cyclotomic v-schur A Hecke cyclotomic Hecke permutation endmorphism ring A Hecke cyclotomic Hecke Cherednik category O Rouquier v v-schur Varagnolo-Vasserot L(Λ ) Fock cyclotomic v-schur Uglov higher level Fock Yvonne Shan Cherednik graded representation theory 8 Hecke Hecke Lie Introduction affine Lie Hecke Hecke / Hecke / Hecke Kleshchev Young Lascoux-Leclerc-Thibon Young affine Lie affine graded representation theory Hecke Lascoux-Leclerc-Thibon-A Hecke
54 54 affine / Hecke Kazhdan-Lusztig Kazhdan-Lusztig Specht A Hecke q Schur A Hecke {T w } w W Kazhdan-Lusztig {C w } w W Kazhdan-Lusztig Lascoux-Leclerc-Thibon-affine U v (sl l ) L(Λ ) Fock upper A Hecke Specht coordinate free / [A:book] [Ari96] [ARS] Susumu Ariki, Representations of quantum algebras and combinatorics of Young tableaux, University Lecture Series, 6. American Mathematical Society, Providence, RI,. Susumu Ariki, On the decomposition numbers of the Hecke algebra of G(m,,n)., J. Math. Kyoto Univ. 36 (996), no. 4, Maurice Auslander, Idun Reiten and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge,
55 Hecke [ASS] Ibrahim Assem, Daniel Simson, Andrzej Skowron ski, Elements of the representation theory of associative algebras, vol.-3, Cambridge University Press, 6-7 [BR] Héléne Barcelo, Arun Ram, Combinatorial representation theory, New perspectives in algebraic combinatorics (Berkeley, CA, ), 3 9, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 999. [CG] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 997. [DDPW] Bangming Deng, Jie Du, Brian Parshall, Jianpan Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs, 5. American Mathematical Society, Providence, RI, 8 [FH] William Fulton and Joe Harris, Representation theory. A first course, Graduate Texts in Mathematics, 9. Readings in Mathematics. Springer-Verlag, 99. [Ful] William Fulton, Young tableaux, London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 997. [HK] Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, 4. American Mathematical Society, Providence, RI,. [HTT] Hotta Ryoshi, Takeuchi Kiyoshi, Tanisaki Toshiyuki D-modules, perverse sheaves, and representation theory, Progress in Mathematics, 36. Birkhäuser, 8 [Mat] Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, 5. American Mathematical Society, Providence, RI, 999. [Mac] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Second Edition, 995 [JK] Gordon James and Adalbert Kerber, The representation theory of the symmetric group, encyclopedia of Mathematics and its Applications, 6. Addison-Wesley Publishing Co., Reading, Mass., 98 [K:book] Masaki Kashiwara, Bases cristallines des groupes quantiques, Cours Spe cialise s, 9. Socie te Mathe matique de France, Paris,. [Kas9] Masaki Kashiwara, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (99), no., [Kas93] Masaki Kashiwara, Global crystal bases of quantum groups Duke Math. J. 69 (993), no., [Kl:book] Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, 63. Cambridge University Press, Cambridge, 5. [Kle] Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, arxiv:math/rt [LLT96] Alain Lascoux, Bernard Leclerc and J.Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 8 (996), no., [Lus] George Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (99), no., [L:book] George Lusztig, Introduction to quantum groups, Progress in Mathematics,. Birkhäuser Boston, Inc., Boston, MA, 993. [Sa] Bruce Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric
56 56 functions, Second edition, Graduate Texts in Mathematics, 3, Springer-Verlag, [Iw],,, 978 [O] (),, 6 [TH],,, 6 [TN],,, 987
平成 15 年度 ( 第 25 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (
1 1.1 X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (X 1,..., X n ) ( ) X 1,..., X n f 1,..., f r A T X + XA XBR 1 B T X + C T QC = O X 1.2 X 1,..., X n X i X j X j X i = 0, P i
More informationλ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T
0 2 8 8 6 3 0 0 Young Young [F] 0.. Young λ n λ n λ = (λ,, λ l ) λ λ 2 λ l λ = ( m, 2 m 2, ) λ = n, l(λ) = l {λ n n 0} P λ = (λ, ), µ = (µ, ) n λ µ k k k λ i µ i λ µ λ = µ k i= i= i < k λ i = µ i λ k >
More informationi Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.
R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................
More informationMacdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona
Macdonald, 2015.9.1 9.2.,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdonald,, q., Heckman Opdam q,, Macdonald., 1 ,,. Macdonald,
More information2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c
GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,
More informationMazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ
Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R
More information. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2
2014 6 30. 2014 3 1 6 (Hopf algebra) (group) Andruskiewitsch-Santos [AFS09] 1980 Drinfeld (quantum group) Lie Lie (ribbon Hopf algebra) (ribbon category) Turaev [Tur94] Kassel [Kas95] (PD) x12005i@math.nagoya-u.ac.jp
More information数学Ⅱ演習(足助・09夏)
II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w
More information2010 ( )
2010 (2010 1 8 2010 1 13 ( 1 29 ( 17:00 2 3 ( e-mail (1 3 (2 (3 (1 (4 2010 1 2 3 4 5 6 7 8 9 10 11 Hesselholt, Lars 12 13 i 1 ( 2 3 Cohen-Macaulay Auslander-Reiten [1] [2] 5 [1], :,, 2002 [2] I Assem,
More informationDynkin Serre Weyl
Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................
More information2.3. p(n)x n = n=0 i= x = i x x 2 x 3 x..,?. p(n)x n = + x + 2 x x 3 + x + 7 x + x + n=0, n p(n) x n, ( ). p(n) (mother function)., x i = + xi +
( ) : ( ) n, n., = 2+2+,, = 2 + 2 + = 2 + + 2 = + 2 + 2,,,. ( composition.), λ = (2, 2, )... n (partition), λ = (λ, λ 2,..., λ r ), λ λ 2 λ r > 0, r λ i = n i=. r λ, l(λ)., r λ i = n i=, λ, λ., n P n,
More information2016 Course Description of Undergraduate Seminars (2015 12 16 ) 2016 12 16 ( ) 13:00 15:00 12 16 ( ) 1 21 ( ) 1 13 ( ) 17:00 1 14 ( ) 12:00 1 21 ( ) 15:00 1 27 ( ) 13:00 14:00 2 1 ( ) 17:00 2 3 ( ) 12
More information1 G K C 1.1. G K V ρ : G GL(V ) (ρ, V ) G V 1.2. G 2 (ρ, V ), (τ, W ) 2 V, W T : V W τ g T = T ρ g ( g G) V ρ g T W τ g V T W 1.3. G (ρ, V ) V W ρ g W
Naoya Enomoto 2002.9. paper 1 2 2 3 3 6 1 1 G K C 1.1. G K V ρ : G GL(V ) (ρ, V ) G V 1.2. G 2 (ρ, V ), (τ, W ) 2 V, W T : V W τ g T = T ρ g ( g G) V ρ g T W τ g V T W 1.3. G (ρ, V ) V W ρ g W W G- G W
More information0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t
e-mail: koyama@math.keio.ac.jp 0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo type: diffeo universal Teichmuller modular I. I-. Weyl
More informationA11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18
2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1
More informationSAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
More informationX G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
More informationDipper-James 22 7
22 7 1 1 1.1.............................. 1 1.2 Frobenius............................ 8 1.3 GL n (E)...................... 10 1.4 Φ d -torus........................ 13 1.5 Young................... 19
More informationAffine Hecke ( A ) Irreducible representations of affine Hecke algebras (survey talk with emphasis on type A) (Syu Kato) Recently, there are
Affine Hecke ( A ) Irreducible representations of affine Hecke algebras (survey talk with emphasis on type A) (Syu Kato) 20 10 29 Recently, there are several successful attempts on the classification of
More informationコホモロジー的AGT対応とK群類似
AGT K ( ) Encounter with Mathematics October 29, 2016 AGT L. F. Alday, D. Gaiotto, Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010), arxiv:0906.3219.
More informationSiegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo
Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo] 2 Hecke ( ) 0 1n J n =, Γ = Γ n = Sp(n, Z) = {γ GL(2n,
More information等質空間の幾何学入門
2006/12/04 08 tamaru@math.sci.hiroshima-u.ac.jp i, 2006/12/04 08. 2006, 4.,,.,,.,.,.,,.,,,.,.,,.,,,.,. ii 1 1 1.1 :................................... 1 1.2........................................ 2 1.3......................................
More information1 Ricci V, V i, W f : V W f f(v ) = Imf W ( ) f : V 1 V k W 1
1 Ricci V, V i, W f : V W f f(v = Imf W ( f : V 1 V k W 1 {f(v 1,, v k v i V i } W < Imf > < > f W V, V i, W f : U V L(U; V f : V 1 V r W L(V 1,, V r ; W L(V 1,, V r ; W (f + g(v 1,, v r = f(v 1,, v r
More informationK 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X
2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ
More informationKhovanov Lauda Rouquier Categorification,
Khovanov Lauda Rouquer Categorfcaton, , 2011 12 5 9, Khovanov Lauda Rouquer Khovanov Lauda Rouquer Khovanov Lauda Rouquer Seok-Jn Kang, Shunsuke Tsuchoka, Myungho Km, Se-jn Oh ) 2013 12 1 Introducton 1
More information2018 : msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald Weyl Demazure. g, h Cartan., Q := i I Zα i h root lattice, Q + := i I Z 0α
2018 : 2018 21 msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald 1.1. Weyl Demazure. g, h Cartan, Q := i I Zα i h root lattice, Q + := i I Z 0α i Q, P := i I Zϖ i h g weight lattice ;, ϖ i h, i I,
More information1
1 Borel1956 Groupes linéaire algébriques, Ann. of Math. 64 (1956), 20 82. Chevalley1956/58 Sur la classification des groupes de Lie algébriques, Sém. Chevalley 1956/58, E.N.S., Paris. Tits1959 Sur la classification
More informationEinstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
More information平成 30 年度 ( 第 40 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 30 ~8 年月 72 月日開催 30 日 [6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1,
[6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1, B 2, B 3 A i 1 B i+1 A i+1 B i 1 P i i = 1, 2, 3 3 3 P 1, P 2, P 3 1 *1 19 3 27 B 2 P m l (*) l P P l m m 1 P l m + m *1 A N
More information1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac
More information006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
More informationD 24 D D D
5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6
More informationMilnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P
Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo
More information24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
More informationsequentially Cohen Macaulay Herzog Cohen Macaulay 5 unmixed semi-unmixed 2 Semi-unmixed Semi-unmixed G V V (G) V G V G e (G) G F(G) (G) F(G) G dim G G
Semi-unmixed 1 K S K n K[X 1,..., X n ] G G G 2 G V (G) E(G) S G V (G) = {1,..., n} I(G) G S square-free I(G) = (X i X j {i, j} E(G)) I(G) G (edge ideal) 1990 Villarreal [11] S/I(G) Cohen Macaulay G 2005
More information1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi
1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys
More informationII Time-stamp: <05/09/30 17:14:06 waki> ii
II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................
More informationmain.dvi
SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1
More information1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half
Wess-Zumino-Witten 1999 3 18 Wess-Zumino-Witten., Knizhnik-Zamolodchikov-Bernard,,. 1 Affine Lie 2 1.1 Affine Lie.............................. 2 1.2..................................... 3 2 WZW 4 3 Knizhnik-Zamolodchikov-Bernard
More informationk + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+
1 SL 2 (R) γ(z) = az + b cz + d ( ) a b z h, γ = SL c d 2 (R) h 4 N Γ 0 (N) {( ) } a b Γ 0 (N) = SL c d 2 (Z) c 0 mod N θ(z) θ(z) = q n2 q = e 2π 1z, z h n Z Γ 0 (4) j(γ, z) ( ) a b θ(γ(z)) = j(γ, z)θ(z)
More informationK E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................
More information, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n
( ), Jürgen Berndt,.,. 1, CH n.,,. 1.1 ([6]). CH n (n 2), : (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube. (iii). (iv) ruled minimal, equidistant. (v) normally homogeneous submanifold F k tube. (vi) normally
More information( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h
2011 9 5 1 Lie 1 2 2.1 (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h
More information( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv
( ) 1 ([SU] ): F K k Z p - (cf [Iw2] [Iw3] [Iw6]) K F F/K Z p - k /k Weil K K F F p- ( 41) Z p - Weil Weil F F projective smooth C C Jac(C)/F ( ) : 2 3 4 5 Tate Weil 6 7 Z p - 2 [Iw1] 2 21 K k k 1 k K
More informationB [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (
. 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1
More information2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) =
2018/10/04 IV/ IV 1/12 2018 IV/ IV 10 04 * 1 : ( A 441 ) yanagida[at]math.nagoya-u.ac.jp https://www.math.nagoya-u.ac.jp/~yanagida 1 I: (ring)., A 0 A, 1 A. (ring homomorphism).. 1.1 A (ideal) I, ( ) I
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More informationJuly 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i
July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac
More information1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc
013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8
More information講義ノート 物性研究 電子版 Vol.3 No.1, (2013 年 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 10 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K
2 2 T c µ T c 1 1.1 1911 Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 1 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K τ 4.2K σ 58 213 email:takada@issp.u-tokyo.ac.jp 1933 Meissner Ochsenfeld λ = 1 5 cm B = χ B =
More informationE1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1
E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 (4/12) 1 1.. 2. F R C H P n F E n := {((x 0,..., x n ), [v 0 : : v n ]) F n+1 P n F n x i v i = 0 }. i=0 E n P n F P n
More informationSiegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p
Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler
More informationII 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
More informationii
ii iii 1 1 1.1..................................... 1 1.2................................... 3 1.3........................... 4 2 9 2.1.................................. 9 2.2...............................
More informationII R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
More information1 4 1 ( ) ( ) ( ) ( ) () 1 4 2
7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)
More information1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe
3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)
More informationBruhat
SGC - 77 Bruhat ([22]) 3 3.11 2010 4 ii 1 1 1.1... 1 1.2... 5 1.3... 8 1.4 1... 11 1.5 2... 14 2 18 2.1... 18 2.2... 25 2.3... 30 3 36 3.1... 36 3.2... 42 3.3... 49 3.3.1... 49 3.3.2... 50 3.3.3... 52
More informationA 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.
A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U,
More information数学メモアール 第4巻, (2004)
6 9 7 i 2002 5 6 0, [TUY] Lie, P ŝl 2, 3 ) OPE) 2) 3) factorization property ), 2, 4 2) 4, 3) 7, 2, OPE 3, P n 3, 4, 5 6, 7 6, 7 factorization property,,, Lie Lie 2, [K] 5.2, 7.9 5.2 3.4, 7.9, 6.2, 6,
More informationQCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann
More information,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
More information[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2
On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields (Yoichi Mieda) Graduate School of Mathematical Sciences, The University of Tokyo 0 l Galois K F F q l q K, F K,
More informationver Web
ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3
More informationii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
More informationnote4.dvi
10 016 6 0 4 (quantum wire) 4.1 4.1.1.6.1, 4.1(a) V Q N dep ( ) 4.1(b) w σ E z (d) E z (d) = σ [ ( ) ( )] x w/ x+w/ π+arctan arctan πǫǫ 0 d d (4.1) à ƒq [ƒg w ó R w d V( x) QŽŸŒ³ džq x (a) (b) 4.1 (a)
More informationS I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
More information(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law
I (Radom Walks ad Percolatios) 3 4 7 ( -2 ) (Preface),.,,,...,,.,,,,.,.,,.,,. (,.) (Basic of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................
More information2/14 2 () (O O) O O (O O) id γ γ id O O γ O O O γ η id id η I O O O O I γ O. O(n) n *5 γ η γ S M, N M N (M N)(n) ( ) M(k) Sk Ind S n S i1 S ik N(i 1 )
1/14 * 1. Vassiliev Hopf P = k P k Kontsevich Bar-Natan P k (g,n) k=g 1+n, n>0, g 0 H 1 g ( S 1 H F(Com) ) ((g, n)) Sn. Com F Feynman ()S 1 H S n ()Kontsevich ( - - Lie ) 1 *2 () [LV12] Koszul 1.1 S F
More informationcompact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1
014 5 4 compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) 1 1.1. a, Σ a {0} a 3 1 (1) a = span(σ). () α, β Σ s α β := β α,β α α Σ. (3) α, β
More informationt = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
More information4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
More information2
III ( Dirac ) ( ) ( ) 2001. 9.22 2 1 2 1.1... 3 1.2... 3 1.3 G P... 5 2 5 2.1... 6 2.2... 6 2.3 G P... 7 2.4... 7 3 8 3.1... 8 3.2... 9 3.3... 10 3.4... 11 3.5... 12 4 Dirac 13 4.1 Spin... 13 4.2 Spin
More informationmeiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
More informationTwist knot orbifold Chern-Simons
Twist knot orbifold Chern-Simons 1 3 M π F : F (M) M ω = {ω ij }, Ω = {Ω ij }, cs := 1 4π 2 (ω 12 ω 13 ω 23 + ω 12 Ω 12 + ω 13 Ω 13 + ω 23 Ω 23 ) M Chern-Simons., S. Chern J. Simons, F (M) Pontrjagin 2.,
More information211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
More information) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
More informationuntitled
Lie L ( Introduction L Rankin-Selberg, Hecke L (,,, Rankin, Selberg L (GL( GL( L, L. Rankin-Selberg, Fourier, (=Fourier (= Basic identity.,,.,, L.,,,,., ( Lie G (=G, G.., 5, Sp(, R,. L., GL(n, R Whittaker
More informationnewmain.dvi
数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
More information20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
More information2016
2016 1 G x x G d G (x) 1 ( ) G d G (x) = 2 E(G). x V (G) 2 ( ) 1.1 1: n m on-off ( 1 ) off on 1: on-off ( on ) G v v N(v) on-off G S V (G) N(v) S { 3 G v S v S G G = 1 OK ( ) G 2 3.1 u S u u u 1 G u S
More informationA = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
More information30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
More informationd ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d )
23 M R M ϕ : R M M ϕt, x) ϕ t x) ϕ s ϕ t ϕ s+t, ϕ 0 id M M ϕ t M ξ ξ ϕ t d ϕ tx) ξϕ t x)) U, x 1,...,x n )) ϕ t x) ϕ 1) t x),...,ϕ n) t x)), ξx) ξ i x) d ϕi) t x) ξ i ϕ t x)) M f ϕ t f)x) f ϕ t )x) fϕ
More information( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1
2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h
More information第5章 偏微分方程式の境界値問題
October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ
More information1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,.
1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, 2015. webpage,.,,. 2 1 (1),, ( ). (2),,. (3),.,, : Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the
More information1 Slodowy 005 3 9 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 A n 5 5 5... 5 5 5 D n 5 5 5... 5 5 5 E 6 5 5 5 5 5 5 E 7 5 5 5 5 5 5 5 E 8 5
More information?
240-8501 79-2 Email: nakamoto@ynu.ac.jp 1 3 1.1...................................... 3 1.2?................................. 6 1.3..................................... 8 1.4.......................................
More informationn ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
More information非可換Lubin-Tate理論の一般化に向けて
Lubin-Tate 2012 9 18 ( ) Lubin-Tate 2012 9 18 1 / 27 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 Lubin-Tate GL n n 1 Lubin-Tate ( ) Lubin-Tate 2012
More information医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
More informationI. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]
More information. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
More informationI , : ~/math/functional-analysis/functional-analysis-1.tex
I 1 2004 8 16, 2017 4 30 1 : ~/math/functional-analysis/functional-analysis-1.tex 1 3 1.1................................... 3 1.2................................... 3 1.3.....................................
More information2 TOMOYUKI ARAKAWA 2. Beilinson-Drinfeld W W. Weyl. g C Lie, G, W Weyl, h Cartan. S(h) W S(h) W. S(h) 3 Heisenberg( ) (free boson). Fateev-Lukyanov [F
PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW TOMOYUKI ARAKAWA ( ). Borcherds [Bor86] (vertex algebra),,. W. W Virasoro ([KRW03]),,. W Zamolodchikov[Zam85]. Feigin-Frenkel[FF90], Kac-Roan-Wakimoto[KRW03],
More informationV(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
More information16 B
16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..
More informationNote.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
More informationPart () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
More information