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
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1 Naoya Enomoto paper
2 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 V 1.4. (1) G V W V/W G ρ ρ : G GL(V/W ); g ρ g ρ g ( v) = ρ g (v) (2) G 2 (ρ 1, V 1 ), (ρ 2, V 2 ) ρ 1 ρ 2 : G GL(V 1 V 2 ); g ρ 1 (g) ρ 2 (g) ρ 1 (g) ρ 2 (g)(v 1, v 2 ) = (ρ 1 (g)(v 1 ), ρ 2 (g)(v 2 )) 1.5. G V G-V 1.6. G (ρ, V ) (τ, W ) 3 (1) V V G GL(V ); g (ρ ) g ρ g(f)(v) = ρ(f(g 1 v)) 2
3 (ρ, V ) (2) V W V W G GL(V W ); g (ρ τ) g (ρ τ) g (v w) = ρ g (v) τ g (w) (ρ τ, V W ) (3) V W Hom(V, W ) g ϕ(v) = g (ϕ(g 1 v)) G Hom(V, W ) G V g 1 ϕ W g V g ϕ W V W V W (1),(2),(3) (3) V W Hom(V, W ) V W f w ϕ f,w ϕ f,w (v) = f(v)w V v i v i v V f(v)w = 0 f = i a iv i v = v j 0 = f(v j )w = a j w w = 0 a i = 0 f = 0 f w = 0 G- ϕ (g f) (g w) (v) = (g f)(v)(g w) = f(g 1 v)(g w) (g ϕ f w )(v) = g (ϕ f w (g 1 v)) = g (f(g 1 v)w) = f(g 1 v)(g w) 1.7. G X V = e x x X G ( ) g a x e x = a x e g x x X x X G X G G G R 2 paper C C Schur 3
4 2.1. (ρ, V ) G W V G- V = W W W G- W V = W U W U p : V W p 0 = 1 σ G ρ σ p ρ 1 σ p : V W p(w ) W p 0 : V W W G-ρ 1 σ w W p ρ 1 σ (w) = ρ 1 σ p 0 (w) = 1 σ G ρ σ ρ 1 σ (w) = w p 0 V W W ρ τ p 0 ρ 1 τ = 1 σ G W = v V p 0 (v) = 0} ρ τ ρ σ p ρ 1 σ ρ 1 τ = 1 ρ τ p 0 = p 0 ρ τ w W p 0 ρ τ (w ) = ρ τ p 0 (w ) = 0 σ G ρ τσ p ρ 1 τσ = p 0 ρ τ (v) W W G G C Remark / K K G 2.1 key word 2.4. C V G-V, x, y V ρ g x, ρ g y = x, y V G- (, ) x, y := 1 (ρ g x, ρ g y) V G- 4
5 2.5. G (ρ, V ) V G-V G- W W G-G Schur 2.6. Schur V, W G ϕ : V W G- (1) ϕ 0- (2) V = W λ C ϕ = λ I I (1) ϕ : V W Ker ϕ, Im ϕ V, W G-V Ker ϕ = 0} V 0} 0-V ϕ Im ϕ = 0} W V 0} Im ϕ = W (2) C ϕ λ C v 0 (ϕ λ I)v = 0 Ker(ϕ λ I) 0}V ϕ λ I = 0 ϕ = λ I 2.7. G 1 (ρ, V ) G G g G ρ g : V V G- Schur ρ g = λ g I V ρ g 1 G 1 Remark V T i } V T i } T i } G V ρ g } V G ρ g ρ g } V 1 G 1 G G 1 ρ g } ρ g } G G G V V = U i U i V = V (V k ) V (V k ) = Ui =Vk U i V V V k 5
6 V = U i = W j 2 G V k } V (V k ) = Ui =Vk U i, V (V k ) = Wj =Vk W j V = k V (V k ) = k V (V k ) U i, W j V (V 1 ) = U 1 U p, V (V 1 ) = W 1 W q π j : V W j U i π j Ui = θ ij 1 i p j > q θ j 0 G- Schur 0- U i W j 1 i p, j > q 0- V u i U i u W j u U i u = π 1 (u) + π 2 (u) + π q (u) + π q+1 (u) + u = θ i1 (u) + θ i2 u + + θ iq u + θ i,q+1 (u) + θ ij = 0 (1 i p, j > q) u = θ i1 (u) + θ i2 u + + θ iq u + θ i,q+1 (u) V (V 1 ) 1 i p U i V (V 1 )V (V 1 ) = p i=1 U i V (V 1 ) V (V 1 ) = V (V 1 )V (V k ) = V (V k ) V 3 G G (ρ, V ) χ V : G C χ V (g) = tr(ρ g ) (ρ, V ) 3.2. G (ρ, V ) χ V (1) χ V G (2) χ V (g 1 ) = χ V (g) χ V (hgh 1 ) = tr(ρ hgh 1) = tr(ρ h ρ g ρ 1 h ) = tr(ρ g) = χ V (g) ρ g λ i ρ g 1 λ i = λ 1 i χ V (g) = i λ i = i λ i 1 = tr(ρ 1 g ) = χ V (g 1 ) 6
7 3.3. G G V χ V G 3.4. V, W G (1) χ V W = χ V + χ W (2) χ V W = χ V χ W (3) χ V = χ V G (ρ, U) U G = v V ρ g (v) = v} ϕ := 1 ρ g End (U) 3.5. ϕ : U U G U Im(ϕ) u = ϕ(v) ( ) 1 ρ h (u) = ρ h ρ g (v) = 1 ρ hg (v) = 1 ρ g (v) = ϕ(v) = u Im(ϕ) U G u U G ϕ(u) = 1 ρ g (u) = 1 u = u U G Im ϕ 3.6. dim(u G ) = 1 χ V (g) U G U dim(u G ) = tr(ϕ) 3.7. dim(u G ) = tr(ϕ) = 1 tr ρ g = 1 χ V (g) Hom(V, W ) G = f : V W : G- } 7
8 Hom(V, W ) G g ϕ(v) = g (ϕ(g 1 v)) g ϕ = ϕ G ρ g ϕ = ϕ ρ g G- g ϕ = ϕ V, W Schur dim(hom(v, W ) G 1 V ) = = G W 0 V = W 1 χ Hom(V,W ) (g) = dim(hom(v, W ) G ) = Hom(V, W ) = G V W 1 V = G W 0 V = W χ Hom(V,W ) (g) = χ V (g) χ W (g) = χ V (g) χ W (g) 3.8. V, W G χ V, χ W = 1 χ V (g)χ W (g) = G a, b : G C a, b = 1 b(g)a(g) G 1 V = G W 0 V = W 3.9. (1) (2) G V χ V, χ V = 1 (3) V V i V i a i = χ V, χ Vi (4) G (5) = V i (dim(v i)) 2 (1) (2) G V V V = V a i i χ V = V a i iχ Vi χ V, χ V = i a2 i 1 (3) (4) G R χ R (g) g e 0g = e χ R (g) = R R = i V a i i a i = χ Vi, χ R = 1 (χ V i (e) ) = dim(v i ) (5) = dim(r) = i (dim(v i)) 2 8
9 G a : G C G (ρ, V ) ϕ a,v = a(g) ρ g : V V G G- a G a G ϕ a,v (ρ h (v)) = a(g)ρ g (ρ h (v)) = a(hgh 1 )ρ hgh 1(ρ h (v)) ( g hgh 1 ) = ρ h ( a(hgh 1 )ρ g (v) = ρ h ( a(g)ρ g (v) = ρ h ϕ a,v (v) ) ) ( a ) G- a h 1, h 2, h G h 1 h 1 h = h 2 a(h 1 ) a(h 2 ) G R ρ h ϕ a,v (e 1 h ) = ρ h( a(g)ρ g (e 1 h )) a G = a(g)e hgh 1 = a(h 1 gh)e g ( hgh 1 g ) ϕ a,v ρ h (e 1 h ) = a(g)ρ g (e 1 ) = a(g)e g e g g G} α(h 1 ) α(h 1 h 1 h) = α(h 2 ) ϕ a,v G G G G a 0 a 9
10 a ϕ a,v G- Schur ϕ a,v = λ I V n = dim(v ) nλ = tr(ϕ a,v ) = a(g) tr(ρ g ) = a(g)χ V (g) = a, χ V ( χ V (g) = χ V (g)) = 0 n 0 λ = 0 V V G R 0 = ϕ a,v (e 1 ) = a(g)ρ g (e 1 ) = a(g)e g e g g G} g G a(g) = G G (1) χ 1, χ 2 G χ 1 (g)χ 2 (g 1 ) = (2) X(G) G χ X(G) χ(g)χ(h 1 ) = (χ 1 = χ2 ) 0 (χ 1 = χ2 ) C G (g) (g h) 0 (g h) (1) (2) f g (h) = 1 (g h) 0 (g h) f g = i λ iχ i λ i = f g, χ i = 1 f g (h)χ i (h) ( ) h G = 1 χ i (h) ( f g (h) g h 1 0) h C(g) = C(g) χ i (g) ( χ i ) 10
11 f g (x) = i λ i χ i (x) = C(g) χ i (g)χ i (x) i χ X(G) χ(g)χ(x) = C(g) f g(x) = C G (g) g x 0 g x [Serre] J.P.Serre Linear Representation of Finite Groups Springer-Verlag [FH] Representation Theory A First Course(Fulton-Harris Springer-Verlag GTM129) 11
Dynkin 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...........................................
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