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2 Frobenius formula Frobenius forumla hook length forumla Giambelli Newton Pieri Littlewood-Richardson rule Kostka Kostka Frobenius forumla involution ω Skew Frobenius Frobenius :Littelwood-Richardson rule :Pieri s rule :Murnaghan-Nakayama rule

3 6.3. Hopf Hopf A d G/H = Weyl Functor

4 Weyl Fulton-Harris [] U(n), SU(n) GL(n, C) SL(n, C) SU(n) Section 3 Section 2 Section 5 [] Littlewood-Ricahrdson rule [2] Macdonald [2] Fulton-Harris [] [] 4

5 . X X X.. X = {, 2,, d} d S d, id, e. i j (i, j) S d i, j {,, d} \ {i, j} i j (i, j) = (j, i) 2. (i, i + ) i, j 3. l i,, i l (i, i 2,, i l ) S d i i 2 i 2 i 3,, i l i l i l i {,, d} \ {i,, i l } l.. σ S 6 ( ) σ = (, 5, 3) (2, 6).2.. S d = d! 2. σ l σ l = (i, j) 2 = 3. ( ) 2 i j d σ = k k 2 k i k j k d σ (i, j) ( ) 2 i j d σ (i, j) = k k 2 k j k i k d 5

6 .2 G G g,, g l G g,, g l, g,, g l {g, g 2,, g l } G G = g,, g l S d.3. S d (, 2), (2, 3),,(d, d) s i = (i, i + ) S d = s,, s d s 2 i = i =,, d s i s i+ s i = s i+ s i s i+ i =,, d 2 (.) s i s j = s j s i i j 2 S d s 2 i = i =,, d S d = s,, s d s i s i+ s i = s i+ s i s i+ i =,, d 2 s i s j = s j s i i j 2 Proof. ( ) σ = S ( ) ( ) (, 2) = ( ) ( ) ( ) (, 2)(2, 3) = (2, 3) = ( ) ( ) (, 2)(2, 3)(3, 4) = ( ) ( ) (, 2)(2, 3)(3, 4)(, 2)(2, 3)(, 2) = = σ = (, 2) (2, 3) (, 2) (3, 4) (2, 3) (, 2) = (, 2)(2, 3)(, 2)(3, 4)(2, 3)(, 2) 6

7 σ (.) S d (.) G d := s,, s d (.) G d s i (i, i + ) S d s i S d.4. G d (s 2, s 3,, s d )s s 2 s k, (0 k d ) G d = Sd G d = Sd G d = (d )! G d G d d = d! G d S d G d d! G d = d! G d = Sd proof of lemma. {s 2,, s d }G d G d G d σ = (s 3,, s d )s 2 s 3 s k, (0 k d ) s i s i+ s i = s i+ s i s i+ s ((s 3,, s d )s 2 s 3 s k )s = (s 3,, s d )s s 2 s s 3 s k =(s 3,, s d )s 2 s s 2 s 3 s k = (s 2, s 3,, s d )s s 2 s 3 s k G d g i G d g s g 2 s g 3 s g s g 2 s g 3 s g s g 2 s g 3 = g (g 2s g 2)g 3 = g s g 2 s s s s g, g G d gs g g gs g = gs (s 3,, s d )s 2 s 3 s k = (g(s 3,, s d ))s s 2 s 3 s k 7

8 .5. S d = (, 2), (, 3),, (, d), S d = (, 2), (, 2,, d) S d Proof. (, 2,, d)(, 2) = (, 2)(2, 3)(3, 4) (d, d)(, 2) = s s 2 s 3 s d s =s s 2 s s 3 s d = s 2 s s 2 s 3 s d (, 2,, d)(, 2)(, 2,, d) = s 2 = (2, 3) s (s s d ) = s 2 s d (, 2)(,, d)(2, 3){(, 2)(,, d)} = (s 2 s 3 s 4 s d )s 2 (s d s 3 s 2 ) =s 3 s 2 s 3 s 4 s d s d s 3 s 2 = s 3 (, 2), (, 2,, d), (, 2,, d) {(, 2), (, 3),, (, d)} (, k)(, k ) (, 3)(, 2) = (, 2, k) (, 2), (, 2,, d) (, 2), (, 3),, (, d).3 s 2 i = B d = s,, s d { si s i+ s i = s i+ s i s i+ i =,, d 2 s i s j = s j s i i j 2 B d B 2 = s B 2 s k k Z Z {,, d} B 3 s s 2 s s 2 s. 8

9 2 3 s s 2 s s 2 s 2 3 B d s i s i s i s i s i s i = i i+ i i+ i i+ i i+ i s i - i+ i s i i+ i i+ i i+ {,, d} {,, d} s,, s d, s,, s d B d.6. s i s i+ s i = s i+ s i s i+ s i s j = s j s i i j 2 s 2 i s 2 i = S d s i = s i 9

10 ( ) 2 3 σ = = (2, 3) 3 2 σ σ = (, 2)(2, 3)(, 2)(2, 3)(, 2) ( ) 2 n σ = i i 2 i n,, n k i k k σ S d {s i } i i, i +, i + 2 0

11 i i+ i i+ i i+ j j+ i i i+ i+ i+2 i i i+ i+ i+2 i i i+ i+ j j j+ j+ i i+ i+2 i i+ i+2 i i+ j j+.7. (, 2,, d, d) = (, 2)(2, 3) (d, d) σ = s i s ik σ = s j s jl σ s i s ik G d S d.8..4 S d.2. G g, g G g g g = xg x x G G S d σ S d X = {,, d} σ k X X σ l (k) = k l k {k, σ(k),, σ l (k)} X σ = ( {, 4, 7, 2}, {2, 0}, {3}, {5, 9}, {6}, {8, 3, 5}, {, 4} )

12 σ = (, 7, 4, 2)(8, 3, 5)(2, 0)(5, 9)(, 4)(3)(6) (3) 3 3 (k) = 4, 3, 2, 2, 2,, [4, 3, 2, 2, 2,, ] σ [,, 2, 2, 2, 3, 4] 5 5 = σ 0 = (, 2, 3, 4)(5, 6, 7)(8, 9)(0, )(2, 3)(4)(5) σ 0 σ ( ) τ = σ = τσ 0 τ [σ 0 ] = [σ] Proof. σ S d σ(k, k 2,, k l )σ = (σ(k ), σ(k 2 ), σ(k l )) τσ 0 τ = τ(, 2, 3, 4)τ τ(5, 6, 7)τ τ(5)τ = σ S d [d] (d )! = d!/d 2,, d k 2,, k d (, k 2,, k d ) [d] [d] (d )! k i k d = d k= ki k [d,, d, d,, d,, k,, k,,,, }{{}}{{}}{{}}{{} i d i k i d d! {d i d id!}{(d ) i d id!} {k i k ik!} { i i!} i ] 2

13 Proof. [4, 3, 2, 2, 2,, ] (k, k 2, k 3, k 4 )(k 5, k 6, k 7 )(k 8, k 9 )(k 0, k )(k 2, k 3 )(k 4 )(k 5 ) {,, d} 5! 4 (k, k 2, k 3, k 4 ) (k 2, k 3, k 4, k ) (k 5, k 6, k 7 ) 2 (k 8, k 9 )(k 0, k )(k 2, k 3 ) (k 8, k 9 )(k 0, k )(k 2, k 3 ) = (k 0, k )(k 8, k 9 )(k 2, k 3 ) 3 3! [4, 3, 2, 2, 2,, ] 5! (4!)(3!)(2 3 3!)( 2!).9 ( ). S d d d p(d) d d d d = λ + + λ d λ λ 2 λ d 0 i = [d,, d, d,, d,, k,, k,,,, }{{}}{{}}{{}}{{} i d i k C i C i = i d i ] d! {d i d id!}{(d ) i d id!} {k i k ik!} { i i!}. i d i d d! = i d C i d! = λ d (dim V λ) 2 V λ λ S d d 2 λ C λ 3

14 .5 S d.3 ( ). σ S d { σ sgn(σ) = σ well-defined Proof. sgn : S d Z 2 = {±} s i = (i, i+) ϵ i = sgn(s i ) = ϵ i (.) sgn S d s i (i, j) = (i, i + ) (j 2, j )(j, j)(j 2, j ) (i +, i + 2)(i, i + ) sgn((i, j)) = sgn : S d Z 2 well-defined (i, j) s i sgn : S d Z l ( ) l.4 ( ). ker sgn A d d S d = A d (, 2)A d A d = d!/2 Section 7 4

15 2 Section 5 d S d d C[x] = C[x,, x d ] (x,, x d ) σ(x,, x d ) = (x σ (),, x σ (d)) x f(x) = f(x,, x d ) (σf)(x) = f(σ x) Section 3 f(x) C[x] (σf)(x,, x d ) = f(x σ(), x σ(2),, x σ(d) ) (σfg)(x) = (σf)(x)(σg)(x) 2. ( ). f(x) C[x] σf = f σ S d C[x] S[x] = S[x,, x d ] 2.2 ( ). {e 0 (x) =, e (x),, e d (x)} d (t x i ) = i= d ( ) d k e d k (x)t k k=0 d ( + x i t) = i= d e k (x)t k k=0 e (x) = x + + x d, e 2 (x) = x x 2 + x 2 x 3 + x d x d = i<j x i x j, e d (x) = x x d 5

16 e r = i <i 2 < <i r d x i x i2 x ir λ = (λ,, λ d ) (Z 0 ) d, M λ (x) = c(λ) σx λ = c(λ) σ S d σ S d σ(x λ x λ d d ) = c(λ) σ s d x λ σ() xλ d σ(d) /c(λ) x λ x λ d d M λ (x) = x α x α d d α (λ,, λ d ) α = (α,, α d ) λ λ d 0 λ P + {M λ λ P + } S[x] Proof. f f = a ν x ν x ν d d (ν, ν 2,, ν d ) a µ x µ x µ d d σ σ(xµ ) µ µ d 0 M µ (x) f(x) a µ M µ (x) a µ x µ x µ d d µ > µ f(x) {M λ λ P + } {M λ λ P + } 2. (). S[x]. 2. C[y,, y d ] f(y,, y d ) f(e (x),, e d (x)) S[x]

17 Proof. λ λ P + µ i = λ i λ i+ ( i d ), µ d = λ d x λ = x λ x λ d d = x µ (x x 2 ) µ2 (x x d ) µ d M λ (x) e (x) µ e 2 (x) µ2 e d (x) µ d x λ λ d f(e (x),, e d (x)) = 0 x d = 0 e d (x) = 0 e i (x,, x d, 0) d (+x i t) = e k (x)t k x d = 0 f(e (x,, x d, 0),, e d (x,, x d, 0), 0) = 0 d f(y,, y d, 0) = 0 f y d f(y) = y d f (y) 0 = f(e,, e d ) = e d f (e,, e d ) f (e,, e d ) = 0 f f f = 0 p k k Z 0 p k = x k + + x k d p 0 = d 2.3 (Newton ). k ( ) j p k j e j = p k e p k + + ( ) k e k p 0 = ( ) k (d k)e k j=0 d ( ) j p k j e j = p k e p k + + ( ) d e d p k d = 0 j=0 d k k d {p,, p d } S[x] 7

18 Proof. subsection 5.2 k = d X diag(x,, x d ) tr X k = p k (x) 0 = (X x i ) = ( ) j X d j e j (x) 0 = d ( ) j p k j e j = 0 j=0 k > d X l k < d k αe k (x) + (e i (x) ( i k ) ) k j=0 ( )j p k j e j x k+ = x k+2 = x d = 0 p 0 = d k j=0 ( )j p k j e j xk+ = =x d =0 = ( ) k (d k)e k (x) αe k (x) + (e i (x) ( i k ) ) x k+ = = x d = 0 α = ( ) k (d k) {e,, e k } {p,, p d } S[x]{e,, e d } {p,, p d } p k (x) = ( ) k+ ke k + (e i (x) ( i k ) ), ( k d) λ = (λ,, λ d ) λ d, λ d, P λ (x) = p λ (x) p λ d d (x) = αeλ e λ d d + {p,, p d } cλ P λ (x) = 0 {e,, e d } λ c λ = 0 λ c λ = 0 {p,, p d } 8

19 2.3 ( ). f (σf) = sgn(σ)f σ S d f(x,, x i,, x j,, x d ) = f(x,, x j,, x i, x n ) f(x,, x i,, x }{{} i,, x n ) = 0 j (x i x j ) = i<j (x i x j ) f(x) = (x)g(x) g(x) g(x) =. V = k=0 V k P V (t) = k (dim V k )t k W = W k, W k = k U j V k j j=0 P W (t) = P U (t)p V (t) 2.4. x = x C[x] P (t) = t = + t + t ( ). C[x,, x d ] = C[x ] d V = C[x,, x d ] P V (t) = ( t )d = ( ) d + k t k k k=0 9

20 2.6 ( ). S[x] = C[e,, e d ] deg(e l ) = l d P S (t) = t l t s ( + t + t 2 + ) t k + t 2 + t 4 + t 2k 2 t s = t k t dk d s l=0 d } + {{ + d } + + } + {{ + } k d k = λ + + λ d s λ l(λ) d k d k d k t s 2.7. d s d s 20

21 3 3. G G G G < 3. ( ). G V ρ : G GL(V ) V G G V ρ(g)v gv 3.. V, W G. V, W G ϕ : V W G G g G ϕ(gv) = gϕ(v) ker ϕ, imϕ, coker ϕ G 2. V W, V W Λ k (V ), S k (V ) G Λ k (V ) V S k (V ) 3. V = Hom(V, C) ρ : G GL(V ) ρ (g)(v ), ρ(g)u = v, u, u V, v V ρ G ρ (g) = t ρ(g ) : V V 4. Hom(V, W ) = V W G ϕ Hom(V, W ) (gϕ)(v) = gϕ(g v) v V 3.2. V W G Hom G (V, W ) Hom(V, W ) G Hom(V, W ) G Proof. ϕ Hom(V, W ) (gϕ)(v) = gϕ(g v) ϕ ϕ(gv) = gϕ(v) g G G 3.3 ( ). V, W G G Φ : V W V W G Φ intertwining 2

22 3.2. G V, W G V, W Φ V, W 3.3. C ρ(g) = id 3.4. X G G Aut(X) Aut(X) X X {x x X} V g( x X a x x) = a x gx, a x C 3.5. G G G Aut(G) g( a h h) = a h (gh), a h C R G R G R G G C(G) G G (gα)(h) = α(g h) Proof. G G R G G f G h G f(h) f h G { h = h e h (h ) = 0 h h f f = f(h)e h f = f(h)e h f(h)h R G G g( f(h)h) = f(h)gh = f(g h)h gf = f(g h)e h (gf)(h) := f(g h) 22

23 3.6. L(g)( a h h) = a h hg Φ : a h h a h h Φ(R(g)( a h h)) = Φ( a h gh) = a h (gh) = a h h g = L(g)(Φ( a h h)) 3.2 G 3.7. G V G W W W G V = W W G GW W Proof. V H 0 G H(v, w) = g G H 0 (gv, gw) W H W G G V (gv, gw) = (v, w) ρ : G U(V ) GL(V ) 3.4. G G G 3.8 ( ). G 3.9. R C 2 ( ) a a 0 {(x, 0) x C} G 23

24 3.0. Schur s lemma C 3. (Schur s lemma). V, W G ϕ : V W G. ϕ ϕ = 0 2. V = W ϕ = λid λ C Proof. ker ϕ, imϕ V, W C ϕ λ ϕ λid kernel ϕ λid ϕ λid = G V V = V a V a k k V i V k a i V i Proof. V V = W b j j id : V a i i W b j j G V Schur s lemma id V V = W id V a W b a = b 3.3. V = V a i i V i V = V i V i diag(v ) = {(a, a) V i V i a V i } V i subsection G V k V, V V, S k (V ) 24

25 G V G g G ρ(g) : V V G G g λ g id V W g G λ g W V V λ g λ g = λ gg ρ : G C G G = Z ki Z l k i G = Z ki G = Z p l = 0,,, p ρ l : G = Z p k exp (2πk)l p C G = Z p Z p ρ l,l : G (k, k ) exp( 2πkl p + 2πk l ) C p S 3 S 3 U U U gv = sgn(g)v 3 C 3 e, e 2, e 3 ge i = e g(i) (z, z 2, z 3 ) g( z i e i ) = z i e g(i) = z g (i)e i 25

26 g(z, z 2, z 3 ) = (z g (), z g (2), z g (3)) 2 V = {(z, z 2, z 3 ) C z + z 2 + z 3 = 0} S 3 S 3 Proof. V L V L = {(z, z 2, z 3 ) az + bz 2 + cz 3 = 0, z + z 2 + z 3 = 0} S 3 a = b = c = 0 S 3 subsection τ = (, 2, 3) S 3 A 3 = τ = Z 3 S 3 A 3 W S 3 W A 3 τ v i λ i v i = τ 3 v i = λ 3 i v i λ i = ω a i a i Z ω = exp( 2 3 πi) W = V i, V i = Cv i, τv i = ω a i v i, σ = (, 2) S 3 S 3 = σ, τ στσ = τ 2 τ(σ(v i )) = στ 2 (v i ) = ω 2a i σ(v i ) σ(v i ) τ ω 2a i. ω ai ω 2ai ω a i v i, σ(v i ) V i = C{v i, σ(v i )} W 2 S 3 V α = (ω,, ω 2 ), β = (, ω, ω 2 ) C 3 26

27 τα = ωα, τβ = ω 2 β, σα = β, σβ = α V i V G ϕ : V i V 2. ω a i = τv i = v i σ(v i ) v i (a) C(v i ) W S 3 σ(v i ) = v i σ(v i ) = v i (b) C(v i + σ(v i )) C(v i σ(v i )) C(v i + σ(v i )) C(v i σ(v i )) 3.4. S 3 U U V W W = U a U b V c c τ = (, 2, 3) ω ω 2 a + c σ = (, 2) b + c σ a + b τ 3.5. V V V α, β V α α, α β, β α, β β τ ω 2,,, ω σ(α α) = β β σ(α β) = β α V = C{α α, β β}, U = C(α β + β α), U = C(α β β α) V V = U U V G G subsection 27

28 3.4 G subsection subsection σ, τ g G {λ i } g k {λ k } λ k i k =, g λ k i cf. Section 2 g G ρ(g) 3.5. V G G χ V χ V (g) := tr (g) χ V (hgh ) = χ V (g) G class function 3.6. (Ad g f)(h) = f(g hg) 3.7. V, W G χ V W = χ V +χ W, χ V W = χ V χ W, χ V = χ V, χ Λ 2 V (g) = 2 [χ V (g) 2 χ V (g 2 )] Proof. g G order n g n = g V {λ i } V {λ i } i λ n i = λ i = = λ i χ V = χ V λ i Λ 2 V {λ i λ j i < j} i<j λ i λ j = ( λ i ) 2 λ 2 i 2 χ Λ 2 V (g) = 2 [χ V (g) 2 χ V (g 2 )] 28

29 3.8. g {λ i } i λ i = 3.9. χ S 2 (V )(g) = 2 [χ V (g) 2 + χ V (g 2 )] χ Λ k (V )(g) g {λ i } i {λ i λ ik i < < i k n} k tr Λ k (V )(g) = e k = λ i λ ik i < <i k n p k = λ k + λ k n k ( ) j p k j e j = ( ) k (n k)e k j=0 k ( ) j χ V (g k j )χ Λ j (V )(g) = ( ) k (n k)χ Λ k (V )(g) j=0 χ V (g 2 ) χ V (g)χ V (g) + nχ Λ 2 (V )(g) = (n 2)χ Λ 2 (V )(g) χ V (g 3 ) χ V (g 2 )χ V (g) + χ V (g)χ Λ 2 (V )(g) nχ Λ 3 (V )(g) = (n 3)χ Λ 3 (V )(g) χ Λ 3 (V )(g) = 6 (2χ V (g 3 ) 3χ V (g 2 )χ V (g) + χ V (g)χ V (g) 2 ) e d = section 5 i +2i 2 + +di d =d ( ) d k= (i k ) i! i i2!2 i 2 i3!3 i 3 id!d i d 29 p i p i 2 2 p i d d

30 3.2. G X χ V (g) g X fixed-point formula Proof. g G x X g x l l {x, gx, g 2 x,, g l x} g l x = x C{e x,, e g x} l g g g = ge x = e x g V tr g X character table Section S S 3 σ = (, 2) τ = (, 2, 3) U U V 2 0 = ω + ω 2 χ U, χ U, χ V S 3 W W = U a U b V c χ W = aχ U + bχ U + cχ V V V (χ V ) 2 (4, 0, ) (4, 0, ) = (2, 0, ) + (,, ) + (,, ) V V = V U U 30

31 V G G V G = {v V gv = v, g G} V 3.22 ( ). G V End(V ) ϕ = g End(V ). G g G ϕ V V G Proof. ϕ G h G g = hgh h G hϕ(w) = ϕ(w) hϕ(w) = G g G hgw = g w = ϕ(w) G g G ϕ(v ) V G v V G ϕ(v) = G g G gv = G v = v V G imϕ ϕ : V V G ϕ(ϕ(w)) = ϕ(w) ϕ 2 = ϕ ϕ m m = dim V G = tr (ϕ) = G g G g G tr (g) = G χ V (g) G V S = k Sk (V ) S k (V ) V k G S G g G P S G(t) = k dim S k (V ) G t k, 3

32 g V ρ(g) λ,, λ n S k (V ) P S G(t) = k = { G g k {λ i λ in n i + + i n = k} dim(s k (V ) G )t k = k i + +i n =k ( G g G λ (g) i λ n (g) i n t k } tr k (g))t k n det(id ρ(g)t) = λ i (g)t i= = ( + λ t + λ 2 t 2 + ) ( + λ n t + λ 2 nt 2 + ) = k i + +i n=k λ (g) i λ n (g) in t k 3.25 (Molien ). G V S G = S k (V ) G P S G(t) = G det(id ρ(g)t) g G V, W G Hom(V, W ) Hom(V, W ) G = Hom G (V, W ) { if V = W dim Hom G (V, W ) = 0 if V W χhom(v,w ) (g) = χv G G W (g) = χv (g)χ W (g) G { if V = W = dim Hom(V, W ) G = dim G Hom(V, W ) = 0 if V W 32

33 3.7. C class (G) := {class functions on G} G C C(G) (α, β) := α(g)β(g) G 3.26 (). V, W G { (χ V, χ W ) = if V = W χ V (g)χ W (g) = G 0 if V W g G V (χ V, χ V ) = 3. V V i a i a i = (χ V, χ Vi ) (χ V, χ V ) = a 2 i Proof. V = a V + a k V k χ V = a i χ Vi χ Vi 3.6 G R G g( e) G R G e G G R G { 0 g e χ R (g) = G = dim R G g = e 33

34 (χ Vi, χ R ) = χvi (g)χ R (g) = G G χ V i (e) G = dim V i R G G V i dim V i dim R G = G = χ R (e) = (χ R, χ R ) = i dim V 2 i χ R = (dim V i )χ Vi g e 0 = χ R (g) = i (dim V i )χ Vi (g) G 3.3. α : G C G V ϕ α,v = g α(g)g : V V ϕ α,v V G α Proof. α ϕ α,v (hv) = α(g)g(hv) = α(hgh )(hgh )(hv) = α(hgh )(hgh )(hv) g hgh g =h g α(hgh )gv = h g α(g)gv = hϕ α,v (v) ϕ α,v G G f(h) R G G α(g)f(h g h ) = α(g)(ghf)(h ) = ϕ α,r (hf)(h ) =h(ϕ α,r f)(h ) = ( α(g)gf)(h h ) = α(g)f(g h h ) = α(h gh)f(h g h ) 34

35 g h gh (α(g) α(h gh))f(h g h ) = 0 g f(h g 0 ) = (α(g) α(h gh))f(h g ) = α(g 0 ) α(h g 0 h) = 0 g f G g, h G α(g) = α(h gh) α G G C class (G) Proof. α : G C V (α, χ V ) = 0 α C class (G) ϕ α,v : V V α G V ϕ α,v = λ (dim V )λ = tr (ϕ α,v ) = α(g) tr (g) = α(g)χ V (g) = G (α, χ V ) = 0 λ = ϕ α,v = 0 R G ϕ α,r = 0 e R G ϕ α,r α(g)ge = α(g)g = 0 {g} g G R G α(g) = 0 α = 0 C class (G) {χ V } V {χ V } V C class (G) Peter-Weyl C class (G) G dim C class (G) C class (G) [g] f [g] (g ) = { g [g] 0 g / [g] 35

36 3.8. G R(G) G V + W (V W ) ai V i, V i, a i Z χ C : R(G) C C class (G) N N N {C,, C N } {V,, V N } {χ,, χ N } δ ij = (χ i, χ j ) = G g χ i (g)χ j (g) = G C k χ i (C k )χ j (C k ) C k / G C k c ik = G χ i(c k ) ij c ikc jk = δ ij = c ik c jk AA = id A A = id k c kic kj = δ ij δ ij = C i G χ C j k(c i ) G χ Ci C j k(c j ) = χ k (C i )χ k (C j ) G k k 3.34 ( ). g G C(g) χ i (g)χ i (g) = i k G C(g) g, h G C(g) C(h) χ i (g)χ i (h) = 0 i 36

37 3.7 Section 3.5 W G χ W G V G ψ = G χ W (g)g End(V ) V ψ = λid g λ = dim V tr ψ = χw (g)χ V (g) G dim V { dim V = = V dim W = W 0 V W 3.35 (). G V V = i V a i i V V a i i π i := dim V i G χvi (g)g End(V ) V ρ : G GL(V ) G V n n Proof. ϕ W χ V a n = (ϕ, χ n ) V n W n=0 a n t n = G n=0 C C ϕ(c)χ(c) n t n = G C C ϕ(c) χ(c)t C V χ(c) = dim V C = [e] g G χ(g) = tr g g λ,, λ n n = dim V λ i = χ(c) = n i= λ i = n = dim V λ i i λ i = n λ i = r i λ r i R 37

38 λ i = r i = ± χ(c) = n i= ±λ = n g = e a n t n = dim W G (dim V )t + G n=0 C [e] C ϕ(c) χ(c)t dim V t t a n G, G 2 G i V i V V 2 G G 2 (g, g 2 )(v v 2 ) = g v g 2 v 2 V V 2 G V i χ i G G 2 V V 2 χ((g, g 2 )) = χ (g )χ 2 (g 2 ) Proof. A, B tr(a B) = tratrb tr ((g, g 2 )) = tr (g ) tr (g 2 ) V V 2 V V 2 G G 2 G G 2 R(G G 2 ) = R(G ) R(G 2 ) Proof. V V 2 V V 2 (χ, χ) = (χ, χ) = χ((g, g 2 ))χ((g, g 2 )) G G 2 (g,g 2 ) G G 2 = χ (g )χ (g )χ 2 (g 2 )χ 2 (g 2 ) G G 2 g 2 G 2 g G = (χ, χ )χ(g 2 )χ(g 2 ) G 2 g 2 G 2 =(χ, χ )(χ 2, χ 2 ) =. 38

39 V V 2 G G 2 V V 2 dim V V 2 = dim V dim V 2 G G 2 = ( i (dim V i ) 2 )( j (dim V j 2 ) 2 ) = i,j (dim V i dim V j 2 ) 2 = i,j (dim V i V j 2 ) 2 G = G G 2 G = k (dim W k) 2 G G 2 = G (dim V i V j 2 ) 2 = (dim W k ) 2 i,j k V i V j 2 W k V i V j 2 [8] page G V χ V (g) λ C a,, a n Z λ n + a λ n + a 2 λ n a n = 0 leading term 2 π Proof. λ = p/q p, q q ( p q )n + a ( p q )n + + a n = 0 q n p n = q(a p n + a 2 p n 2 q + + a n q n ) p, q Proof. g V µ,, µ N g G = e µ,, µ N (µ i ) G = µ i χ V (g) = µ + + µ N 39

40 3.40. G V G 3.4. Section 3.0 (f f )(g) = f(gh )f (h) g G χ V χ V χ V = G dim V χ V Proof. Section 3.0 V {ρ kl (g)} k,l ρ kl ρ pq = G δ dim V lpρ kq Proof of. χ V χ V = G dim V χ V χ V χ }{{ V = G m } (dim V ) χ m V m+ G m (dim V ) m G m m (dim V ) m G, dim V dim V G 3.9 G V, W G L 0 : V W L(v) = g L 0 (gv) G g G L : V W V W L = 0 V = W L = tr (L 0 )/ dim V 40

41 Proof. L G L(hv) = g L 0 (ghv) = G G g G gh G = G h( g L 0 (gv)) = hl(v). g G (gh ) L 0 (gv) V W L = 0 V = W L = λid λ(dim V ) = tr L = tr (g L 0 g) = tr L 0 G g G 3.43 ( ). V, V G G G ρ ij, ρ kl i, j dim V, k, l dim V. V V (ρ ij, ρ kl) = G ρ ij (g)ρ (g) kl = 0. g G 2. V = V (ρ ij, ρ kl) = G g G ρ ij (g)ρ (g) kl = δ ikδ jl dim V Proof. V V V, V {e i } i, {e k } k L 0 (v) = v, e i e k 0 = L(e j ), e l = ( G = G = G ρ (g) ρ(g)e j, e i )e k, e l g G ρ(g)e j, e i ρ (g) e k, e l = ρ(g)e j, e i e G k, ρ (g)e l g G g G ρ(g)e j, e i ρ (g)e l, e k = (ρ lk, ρ ji ) g G V = V L 0 (v) = v, e i e k tr L 0 = j L 0 (e j ), e j = j e j, e i e k, e j = j δ ji δ kj = δ ki 4

42 δ ki δ jl dim V = tr L 0 dim V δ jl = L(e j ), e l = (ρ ji, ρ lk ) G G dim R G = (dim V i ) 2 i G 3.44 (Peter-Weyl ). G Peter-Weyl Stone-Weierstrass G CG 3.9. G CG group algebra {g g G} C ( g G a g g) ( h G b h h) := gh a g b h gh, a g, b h C C V CG End(V ) 42

43 3.46. G ρ : G GL(V ) ρ : CG End(V ) CG G G CG CG CG G C C(G) convolution f f (f f )(g) := g G f(gh )f (h) G C C(G) C (C(G), ) Proof. f, f f(h)h, f (h)h ( h f(h)h)( k f (k)k) = h,k f(h)f (k)hk = g,k f(gk )f (k)g = g { k f(gk )f (k)}g g G f(gh )f (h). ((f f 2 ) f 3 )(g) = (f f 2 )(gh )f 3 (h) = f (gh k )f 2 (k)f 3 (h) h h,k = f (gk h )f 2 (h)f 3 (k) = f (gk (hk ) )f 2 (hk )f 3 (k) k h k h = f (gh )f 2 (hk )f 3 (k) = (f (f 2 f 3 ))(g). h,k G f f (g) := f(g ) (f ), (f g) = g f f f CG f(g)f f(g )g = f(g)g 43

44 3.49. (C(G), ) = CG = End(W i ) G f f(g)g f(g)ρ i (g). g i g Proof. W i G ρ i : CG End(W i ) (g, g ) G G CG h (g, g )h = ghg CG, End(W i ) f (g, g )f = ρ i (g)fρ i (g ) End(W i ) G G ρ i : CG End(W i ) G G W i EndW i = W i W i G G ρ i W i ϕ : CG End(W i ) CG (dim W i ) 2 dim CG = dim End(W i ) ϕ : CG End(W i ) ϕ(gh) = ϕ(g)ϕ(h) CG = End(W i ) G G End(W i ) End(W i ) CG 3.5. { g α(g)g α C class(g)} CG = C(G) C class (G) C(G) 44

45 Proof. G h( a(g)g)h = a(g)hgh = a(h gh)g = a(g)g a(h gh) = a(g) a(g) W i ρ kl i C(G) End(W i ) C(G) ρ kl i ρ kl i (g)g ϕ j g ρ kl i (g)ρ j (g) j End(W j ) ρ j = ρ pq j Ej pq E pq (p, q) i (g) W i ρ pq i (g) = ρqp i (g ) = ρ pq W i W i ρ kl i j g ρ kl i (g)ρ j (g) = p,q g ρ kl i (g)ρ pq i (g)ei pq = = G dim W i E i kl End(W i ) j G dim W i p,q End(W j ) δ kp δ ql E i pq W i ρ kl i (g) End(W i ) E kl ρ kl i ρ pq i = G δ lp ρ kq i ( dim W i G G E i dim W kl)( E i i dim W i ρ kl i ρ pq i (g) = ρ kl i (gh )ρ pq i (h) = h h s G = h s ρ ks (g) i ρ ls i (h)ρpq i (h) = dim W i s pq) = G dim W i δ lp E i kq. ρ ks i (g)ρ sl i (h )ρ pq i (h) ρ ks i (g)δ lp δ sq = G ρ kq i (g)δ lp dim W i 45

46 2. C class (G) C(G) α, β C class (G), f C(G) α β C class (G), α f = f α α f = f α α(g) = α(hgh ) 3.3 α(g)g CG G α α End(W i ) λ i id Wi End(W i ) End(W i ) 3. χ Wj π j := dim W j G χ Wj (g)g W j Wj χ Wj End(W i ) W j 3.4 χ Wj χ Wj = G dim W j χ Wj π j π j = π j χ Wi χ Wj = 0 i j V, W g χ V χ W = { 0 V = W G dim V V = W CG W CG CGW W W CG CG = W W e G e = w + w W W w, w x CG x = xw + xw W W 46

47 x W x = xw w = w 2 ww = 0 = w w w W CGw CG x xw W w W = CGw G e = w + (e w) (e w) 2 = e + w 2 2w = e w e w CG = CGw + CG(e w) a CG CGa G W W w w = w + w 2, w 2 = w, w 2 2 = w 2, w w 2 = w 2 w = 0 w = w w = w 2 G CG CG a = a g g CG a = a g g CG b a = (ab) w CG w CGw, CGw a, b = ab e (CGw) = CGw ga, b = a, g b G w CG W = CGw E = wcgw E = wcgw = Hom CG (W, W ) = Hom G (W, W ) 47

48 Proof. ϵ, ϵ E = wcgw w ϵϵ E W E W W E E Hom(W, W ) G E (G E) W ((g, ϵ), v) gvϵ W G E E Hom G (W, W ) ϵ E ϕ ϵ Hom G (W, W ) ϕ ϵ (x) = xϵ ϕ ϵ ϕ ϵ (x) = xϵ ϵ = ϕ ϵ ϵ(x) E ϵ ϕ ϵ Hom G (W, W ) ϕ Hom G (W, W ) ϕ(w) = ϵ CGw x W = CGw x = xw ϕ(x) = ϕ(xw) = xϕ(w) = xϵ ϵ = ϕ(w) = wϵ wcgw, ϵ E ϕ(w) = ϵ, ψ(w) = ϵ ϕψ(w) = ϕ(ϵ ) = ϕ(ϵ w) = ϵ ϕ(w) = ϵ ϵ = ψ(w)ϕ(w) Hom G (W, W ) E Hom G (W, W ) = E CG = CG CG CG E = wcg CG CGw = W CG W E = Hom G (W, W ) wcg = W Φ : wcg a h h a h h CGw G gφ( a h h) = a h gh = a h (hg ) = Φ(( a h h)g ) S d Section 4 48

49 3. subsetion CG = End(W i ) 3.54 (S ). S ρ n : S e iθ e inθ C, n Z G Ĝ G Ŝ = Z = {ρ n n Z} (χ m, χ n ) = { 2π (n = m) e inθ e imθ dθ = 2π 0 0 (n m) L 2 (S ) f ˆf(n) = 2π f(θ)e inθ dθ = (χ n, f) 2π 0 {e inθ } n Z L 2 (S ) ˆf : Ĝ n ˆf(n) C Ĝ f(θ) = ˆf(n)e inθ n Z Plancherel 2π 2π 0 f(θ) 2 dθ = n Z ˆf(n) 2 subsection C(G) End(W i ) f g G f(g)ρ i (g), ρ i W i 49

50 Ĝ ρ i f(g) tr (ρ i (g)) = g G G f(g)χ Wi (g)dg C. S Ŝ n f(θ)χ n (θ) dθ S 2π C 3.55 (R ). R ξ R ρ ξ : R x e ixξ U() C R R = {ξ R} = R (χ ξ, χ η ) = e ixξ e ixη dx = δ(ξ η) = 2π R { (ξ = η) 0 (ξ η) R S(R) R S(R) f ˆf(ξ) = f(x)e ixξ dx = (χ ξ, f) S( R) 2π R f(x) = R ˆf(ξ)e ixξ dξ Plancherel f(x) 2 dx = 2π R R ˆf(ξ) 2 dξ 50

51 G C(G) 3.0 ( ). G ρ : G GL(V ) f C(G) End(V ) ˆf(ρ) := g G f(g)ρ(g). V (ρ i, W i ) ˆf(ρ i ) := g G f(g)ρ i (g) End(W i ) ( ρ i, W i ) C(G) f ˆf( ρ i ) = f(g)ρ i (g) End(W i ) i g G 3.49 Ĝ 3. ( ). G f f ˆf(ρ i ) = g G f(g)χ Wi (g) C well-defined C(G) = End(W i ) f f (ρ) = ˆf(ρ) ˆf (ρ). 5

52 3.57. f W χ W (g) { G Wi = W χ W ([W i ]) = 0 W i W χ W (g) = G i χ W ([W i ])χ W i (g) S R Ĝ dim W i dim W i ˆf([Wi ]). i Ĝ g G f(g)χ Wi (g) = i Ĝ χ R (g) f(g) dim W i χ Wi (g) = f(g)χ R (g) = G f(e) g G i g f(g) f h (g) := f(gh) f(h) = f(gh) G g G i = G = G i dim W i k G dim W i χ Wi (g) = G f(k)χ Wi (kh ) = G dim W i tr (ρ i (h ) ˆf(ρ i )) i dim W i f(gh)χ Wi (g) i g G dim W i tr ( i k f(k)ρ i (k)ρ i (h )) ˆf ρ i f (g) = f(g ) f f (e) = f(g)f (g ) = f(g)f(g) = g f(g) 2 f f f f f (e) = G i dim W i tr (ρ i (e) f f (ρ i )) = G 52 dim W i tr ( ˆf(ρ i ) f (ρ i )) i

53 g f(g) 2 = G dim W i tr ( ˆf(ρ i ) f (ρ i )) i C(G) (ρ ij, ρ kl ) = δ ik δ jl / dim V f(h) = G = i = i dim W i i kl dim W i kl dim W i tr (ρ i (h ) ˆf(ρ i )) g g ρ kl i (g)f(g)ρkl i (h) = i ρ kl i (g)f(g)ρ lk i (h ) = i dim W i kl g ρ lk i (g )f(g)ρ lk i (h ) dim W i f(g) tr (ρ i (g)ρ i (h )) g W i 3.2 G H V G H Res G HV H G G V W V H HW W gw = {gw w W } ghw = gw G/H σ gw σw 3.2. V W V = σw σ G/H V {σw } σ G/H V = Ind G HW = IndW V = Ind G HW = σw H W σw H H W G/H G V H 53

54 Proof. V {e σ σ G/H} W = Ce H e σ = σe H V = σ G/H σw 3.6. G R G H R H Proof. R G {e g g G} R H {e h h H} g G e g gr H = σr H G H H W G V V = Ind G HW V IndW = CG CH W G g(g w) = gg w W = W i IndW = IndW i Proof. σ G/H g σ G H G/H e coset σ G/H W W σ W w g σ w W σ V = W σ V v = g σ w σ w σ W g G [gg σ ] = τ G/H gg σ = g τ h h H g τ h H g(g σ w σ ) := g τ (hw σ ) W τ W V g G g g τ = g ρ h g (g(g σ w σ )) = g (g τ (hw σ )) = g ρ (h (hw σ )) (g g)g σ = g (gg σ ) = g g τ h = g ρ h h (g g)(g σ w σ ) = g ρ ((h h)w σ ) = g ρ (h (hw σ )) V = Ind G HW V = Ind G HW V v = g σ w σ w σ W g σ g G gg σ = g τ h g(g σ w σ ) = (gg σ )w σ = (g τ h)w σ = g τ (hw σ ) V Ind G HW W = W i IndW = IndW i 54

55 3.63. G V IndW W = IndW IndW R H = W dim W i i R G = IndR H R G = IndR H = Ind i W dim W i i = i (IndW i ) dim W i H W i H R G V i IndW = V i V j H K G W H Ind G H(W ) = Ind G K(Ind K HW ) Proof. Ind K H = CK CH W Ind G H(W ) = CG CH W = CG CK (CK CH W ) = Ind G K(Ind K HW ) W H G IndW = Hom H (CG, W ) = {f : G W f(gh) = h f(g), h H, g G} F Hom H (CG, W ) (g F )( a g g) = F ( a g gg ) H W (g f)(g) = f(g g) G/H S W = G H W G/H. Γ(S W ) Γ(S W ) = {f : G W f(gh) = h f(g), h H, g G} Proof. F Hom(CG, W ) G (g F )( a g g) = F ( a g gg ) (g F )(h a g g) = F ( a g hgg ) = hf ( a g gg ) = h((g F )( a g g)) g F Hom H (CG, W ) 55

56 F : CG W f(g) := F (g ) f(gh) = F ((gh) ) = h F (g ) = h f(g) f : G W F ( a g g) := a g f(g ) h(f ( a g g)) = a g hf(g ) = a g f(g h ) = a g F (hg) = F (h a g g) IndW = Hom H (CG, W ) σ G/H g σ G F Hom H (CG, W ) Φ : Hom H (CG, W ) F σ G/H g σ F (gσ ) CG CH W = IndW well-defined g σ = g σ h g σ F ((g σ) ) = g σ h F (h gσ ) σ G/H = σ G/H σ G/H g σ hf (h g σ ) = well-defined Φ G g g σ = g τ h σ G/H g σ F (g σ ). Φ((g F )) = g σ (g F )(gσ ) = g σ F (gσ g ) g Φ(F ) = σ g g σ F (gσ ) = τ g τ h F ((g g τ h) ) = τ g τ F (g τ g ) Φ G Φ v = g σ w σ Ψ(v)(hgσ ) := F v (hgσ ) = hw σ F v Hom(CG, W ) g σ g = g σ h g = h gσ g = hgσ F v (h hgσ ) = (h h)w σ = h (hw σ ) = h F v (hgσ ) F v Hom H (CG, W ) ΦΨ = id (ΨΦ(F ))(hg σ ΨΦ = id Φ(Ψ(v)) = g σ F v (g σ ) = g σ w σ, ) = Ψ( g σ F (g ))(hg ) = hf (g ) = F (hg σ 56 σ σ σ )

57 H W T {g σi } σi G/H G/H = n gσ i gg σj / H T (gσ i gg σj ) = 0 g T (gσ gg σ ) T (gσ gg σn )..... T (gσ n gg σ ) T (gσ n gg σn ) IndW g σ i gg σi H gσ i = σ i χ IndW (g) = σ i gg σi ), i such that gσ i = σ i χ W (g 3.67 ( ). C G C H H D,, D r W H χ IndW (C) = G H W r i= D i C χ W (D i ) χ IndW (C) = G/H C C H Proof. c C gcg C c C H c C hch C h H c H hch H C H H C H = D D r C H = g C gσ gg σ C gσ gg σ H T (gσ gg σ ) = 0 χ IndW (C) = 0 C H C H = D D r G/H = {H, g σ H,, g σn H} χ IndW (C) = C g C χ IndW (g) = C g C χ W (gσ i gg σi ), i such that gσ i = σ i 57

58 gσ i gg σi H gσ i gg σi C g σ i gg σi H C g σ i gg σi D k k gσ i gg σi / H gσ i gg σi C gσ i gg σi / D k k χ W D k g C i such that gσ i = σ i gσ i gg σi D k i =,, n, g C n D k = G D K k σ i {gσ i gg σi g C} C g gσ i gg σi C D k = #({g σ i gg σi g C} D k ) i =,, n n χ W (D k ) n D k C g C χ IndW (g) = χ W (gσ C i gg σi ) = G H g C i such that gσ i = σ i C r D k χ W (D k ) W χ W (D k ) = D k = C H χ IndW (C) = G/H C H C W IndW W = W i IndW = IndW i Ind : R(H) W Ind G HW R(G) Res : R(G) V Res G HV R(H) G U H W U IndW = Ind(Res(U) W ) W Ind(Res(U)) = U P P G G/H 58 k=

59 Proof. Φ : U IndW u g σ w σ g σ (g σ u w σ ) Ind(Res(U) W ) G g G [gg σ ] = τ G/H gg σ = g τ h h gφ(u g σ w σ ) = g(g σ (gσ u w σ )) = g τ (hgσ u hw σ ) g(u g σ w σ ) = gu g τ hw σ Φ(g(u g σ w σ )) = Φ(gu g τ hw σ ) = g τ (gτ gu hw σ ) = g τ (hgσ u hw σ ) G W H U G Hom H (W, ResU) = Hom G (IndW, U) H ϕ : W U = Res(U) G ϕ : IndW U W IndW ϕ Proof. IndW = σ G/H σw ϕ Hom H (W, ResU) IndW U σw ϕ ϕ : σw g σ W ϕ U gσ U, ( ϕ(g σ w) = g σ ϕ(w)) [g σ ] = σ G/H g σ G g σ = g σ h ϕ = g σϕ(g σ) ϕ (g σ w) = g σ hϕ((g σ h) g σ w) = g σ hh ϕ(g σ w) = ϕ(g σ w) G gg σ = g τ h ϕ(gg σ w) = g τ ϕ(g τ g τ hw) = g τ ϕ(hw) = g τ hϕ(w) = gg σ ϕ(w) = g ϕ(g σ w) 59

60 ϕ : W U ϕ : IndW U ϕ (g σ w) = g σ ϕ (w) = g σ ϕ(w) ϕ ϕ : W U, ϕ : W U ϕ = ϕ : IndW U W ϕ = ϕ Hom H (W, ResU) Hom G (IndW, U) G Φ : IndW U Φ W : W U Φ 3.70 (Frobenius Resiprocity ). W H U G (χ IndW, χ U ) G = (χ W, χ ResU ) H W U IndW U ResU W Proof. Ind W i = IndW i, Res U i = ResU i W, U dim Hom G (IndW, U) dim Hom H (W, Res(U)) 3.7. S 2 S 3 S 2 U 2 U 2 S 3 U 3, U 3, V 2 ResU 3 = U 2, ResU 3 = U 2, ResV 3 = U 2 U 2 (χ IndU2, χ U3 ) = (χ U2, χ ResU3 ) =, (χ IndU 2, χ U3 ) = (χ U 2, χ ResU3 ) = 0, (χ IndU2, χ U 3 ) = (χ U2, χ ResU 3 ) = 0, (χ IndU 2, χ U 3 ) = (χ U 2, χ ResU 3 ) =, (χ IndU2, χ V3 ) = (χ U2, χ ResV3 ) =, (χ IndU 2, χ V3 ) = (χ U 2, χ ResV3 ) =, IndU 2 = U 3 V 3, IndU 2 = U 3 V G 60

61 3.3 ( ).. G V 0 V = V 0 C G V V G V 0 = V R G 2. G V G J; V V J 2 = id dim C V J V G G B G B Proof. B G V G ϕ : V B V H V G ϕ x V B(x, y) = H(ϕ(x), y) ϕ(x) V well-defined ϕ 2 G V ϕ 2 = λid λ C H(ϕ(x), y) = B(x, y) = B(y, x) = H(ϕ(y), x) = H(x, ϕ(y)) H(ϕ 2 (x), y) = H(ϕ(x), ϕ(y)), H(x, ϕ 2 (y)) = H(ϕ(x), ϕ(y)) λh(x, y) = H(ϕ 2 (x), y) = H(x, ϕ 2 (y)) = λh(x, y) λ = λ λ λh(x, x) = H(ϕ 2 (x), x) = H(ϕ(x), ϕ(x)) λ > 0 H ϕ 2 = id ϕ G ϕ B(x, y) = H(ϕ(x), y) B G λh(x, x) = H(ϕ 2 (x), x) = B(ϕ(x), x) = B(x, ϕ(x)) = H(ϕ(x), ϕ(x)) H ϕ 2 = id ϕ G 6

62 3.73. G V V 0 W 0 V 0 W 0 C V Z n n > 2 R 2 ( ) cos 2πk sin 2πk n n ρ : k sin 2πk n cos 2πk n R 2 R 2 C C V 0 V = V 0 C. V 2. V Proof. V G W V V 0 W {v i } i V 0 (g ij ) g ij R {w i = j v j α j i } W G ij gw i = j,l = p g jl v l α j i = l G ip w p = p,l dim W p= dim V v l ( j= G ip v l α l p = l G ip α l p = dim V j= g jl α j i ) g jl α j i dim W v l ( p= G ip α l p) {G ip } g jl G ip V = V 0 C, W := { w w W } W z(v α) z(v α) W 62

63 g(w i ) = g( v i α j i ) = gv i α j i W W G gvi α j i = G ip w p = p p G ip w p W W W W W {0} W W W V W R := {w + w w W } W W 2 dim C W {w p + w p, iw p + iw p } W R V 0 G G ip = G R ip + G I ip g(w i + w i ) = p G ip w p + G ip w p = p = p G R ipw p + G im ip w p + G R ipw p G I ipw p G R ip(w p + w p ) + G I ip( w p + w p ) W R V 0 W R = V 0 W W = V V G V V = V V = V V G B B ± (x, y) = B(x, y) ± B(y, x) B + (x, y) G B (x, y) G Hom G (V, V ) = Hom G (V, V ) = C Hom G (V, V ) G B +, B V V = V G χ V (g) = χ V (g ) = χ V (g) V V = V 63

64 3.76. V G. χ V V G 2. χ V V G V G 3. χ V V G V G V 0 V G χ V (g 2 ) = V G g G V Proof. χ Λ 2 V (g) = 2 (χ V (g) 2 χ V (g 2 )), χ S 2 V (g) = 2 (χ V (g) 2 + χ V (g 2 )) (χ V, χ V ) = G g G χ V (g)χ V (g) = G χ V (g) 2 Λ 2 V S 2 (V ) g G V, W V W 2. V, W V W 3. V V V 4. V Λ k V k k J, J 2 V, W J J 2 Proof. V V B : (V V ) (V V ) (f v, f 2 v 2 ) f, v 2 f 2, v C G G V V = (V V ) = V V B(x, y) = H(ϕ(x), y) ϕ {e i } i V {e i } i G J J 2 = 64

65 4 Section S d d d d = λ + + λ k λ λ k d see Section 2 p(d) p(d) p(d)t d = d=0 n= t < ( ) = ( + t + t 2 + )( + t 2 + t 4 + )( + t 3 + t 6 + ) t n Proof. t d + t + t 2 + t k + t 2 + t 4 + t 2k 2 t d = t k t 2k2 t lk l l d l } + {{ + } l + + } 2 + {{ + 2 } + } + {{ + } k l k 2 p(d)t d d λ = (λ,, λ k ) i λ i d d λ = (λ,, λ k ) k l(λ) λ d λ,, d 9 9 = k 65

66 λ λ λ 3 λ λ λ (3, 3, 2, ) (4, 3, 2) d subsection S d Section 3 λ + λ λ k T P =P λ = {g S d g T } = S λ S λk, Q =Q λ = {g S d g T } = S µ S µl. µ (, 2, 3)(4, 5) P CS d a λ = g, b λ = sgn(g)g. g P g Q 4.. Section 8 V S d V d g : V d v v d v g () v g (d) V d Proof. w w d = v g () v g (d) g (w w d ) = w g () w g (d) w i = v g (i) v g (g ()) v g (g (d)) 66

67 a λ CS d End(V d ) a λ (V d ) b λ (V d ) S λ V S λ 2 V S λ k V V d Λ µ V Λ µ 2 V Λ µ l V V d (µ,, µ l ) λ Proof. λ. (d) d a λ = g S d g a λ (v v d ) = v v d v v 2 = v v 2 + v 2 v S d (V ) V d Q e S d imaga b λ = V d 2. (,,, ) P = {e} image a λ = V d Q = S d g S d sgn(g)g b λ (v v d ) = v v d 3. λ = P, 2, 3 4, 5, 6 7, 8 9 {, 2, 3} {4, 5, 6}, {7, 8} {9} S 3 S 3 S 2 S a λ S 3 (V V V ) V 6 S 3 (V ) V 6 S 3 (V ) S 3 (V ) S 2 (V ) V b λ v v 2 v 3 v 3 v 2 v (v v 3 ) v 2 Section 8 67

68 S d 4.. λ T a λ, b λ c λ = a λ b λ CS d 4.2. (d) c (d) = a (d) (,, ) c (,,) = b (,,) 4.3 ( ). c λ n λ c 2 λ = n λc λ CS d CS d c λ V λ := (CS d )c λ S d c λ S d d c λ λ λ = (d) c (d) = g V (d) = CS d ( g S d g) = C g S d g g (c (d) ) = c (d) λ = (d) c (d) = g 2. λ = (,, ) c (,,) = sgn(g)g ( a h h)( sgn(g)g) = h a h sgn(h)sgn(g )g = ( a h sgn(h)) sgn(g )g g h g V (,,) = CS d ( g S d sgn(g)g) = C g S d sgn(g)g S d g ( sgn(g)g) = sgn(g ) sgn(g)g 68

69 λ = (,, ) c λ = sgn(g)g 3. S , 2+, 3 V (3) V (,,) 6 = S 3 = dim V 2 i CS 3 V CS 3 = U U V V dim V = 2 (2, ) V (2,) V T 0 a λ = + (, 2), b λ = (, 3) c λ = ( + (, 2))( (, 3)) = + (, 2) (, 3) (, 3, 2) (, 2)c λ = c λ (, 3)c λ = (, 3) + (, 2, 3) (2, 3) (2, 3)c λ = (2, 3) + (, 3, 2) (, 2, 3) (, 2) = c λ (, 3)c λ (, 2, 3)c λ = (, 3)(, 2)c λ = (, 3)c λ (, 3, 2)c λ = (, 3)(2, 3)c λ = (, 3)c λ c λ V (2,) = C{c λ, (, 3)c λ } T 0 (, 3) 2 c λ = ( + (, 3))( (, 2)) = (, 2) + (, 3) (, 2, 3) (, 3)c λ = c λ (, 2)c λ = (, 2) + (, 3, 2) (2, 3) (2, 3)c λ = (2, 3) (, 2, 3) + (, 2, 3) (, 3) = c λ (, 2)c λ (, 2, 3)c λ = (2, 3)(, 3)c λ = c λ (, 2)c λ (, 3, 2)c λ = (, 2)(, 3)c λ = (, 2)c λ V (2,) = C{c λ, (, 2)c λ} 69

70 V (2,) V,2 = 0 T 0, T g = (2, 3) a λ = ga λg, b λ = gb λg, c λ = gc λg P λ = gp λg Q λ = gq λg V λ = CS dc λ = CS dgc λ g = V λ g V λ v vg V λ G V λ = V λ d λ V λ V λ CS d V λ dim V λ λ λ dim V λ d λ T P =P λ = {g S d g T } Q =Q λ = {g S d g T } a λ = g P g, b λ = g Q sgn(g)g A = CS d a = a λ, b = b λ, c = ab = c λ P Q = {e} P Q = {e} S d p q p P, q Q pq = p q p = p, q = q c = ±g g = pq sgn(q) e c p P pa = ap = a. 2. q Q sgn(q)qb = b sgn(q)q = b. 3. p P, q Q pc sgn(q)q = c A c Proof. c = ab = p P,q Q sgn(q)pq pc sgn(q)q = c 70

71 n g g ng pg sgn(q)q = n p gq sgn(q)g g, p, q n pgq = sgn(q)n g n pq = sgn(q)n e g / P Q n g = 0 n g g = sgn(q)n e pq = n e sgn(q)pq = ne c g / P Q t t = p P q = g tg Q g = pgq n g = n pgq = sgn(q )n g = sgn(g tg)n g = sgn(t)n g = n g n g = 0 g / P Q t t = p P q = g tg Q T T = gt T i g(i) t = (i, j) P i jq = g tg = (g (i), g (j)) Q T i, j T i, j T g (i), g (j) T T = gt i, j i j g P Q T T = gt i, j i j T T = gt p P q Q = gqg Q T p T q T 7

72 (, 2)T (4, 2)(, 8)T p P, q Q pt = q T pt = q gt p = q g q = gqg q Q p = (gqg )g = gq g = pq P Q 4.2 ( ). λ, µ λ > µ λ i µ i (3, 3, 2, ) > (3, 3,,, ) λ > µ x A = CG a λ xb µ = 0 = b µ xa λ λ > µ c λ c µ = 0 2. x A c λ xc λ c λ = 0 c λ c λ = n λ c λ n λ C Proof.. x = g S d b µ T gb µ g gt µ λ a λ µ b µ a λ b µ = 0 λ > µ i, j T T λ i = µ i i l λ l > µ l 8, 0 T T

73 t = (i, j) a λ t = a λ, tb µ = b µ a λ b µ = a λ ttb µ = a λ b µ a λ b µ = 0 ta λ = a λ, b µ t = b µ b µ a λ = 0 A x = ag g x = a g g A involution a λ = a λ, b µ = b µ a λ xb µ = 0 b µ x a λ = 0 λ > µ a λ xb µ = 0 x = (b λ a µ ) c λ c µ = a λ (b λ a µ )b µ = 0 2. ()(2) p P, q Q pc λ xc λ sgn(q)q = c λ xc λ (3) c λ xc λ c λ 4.7. λ µ c λ xc µ = 0 x A c λ c µ = 0 Proof. λ > µa λ xb µ = 0x b λ xa µ c λ xc µ = a λ (b λ xa µ )b µ = 0 λ < µ b λ xa µ = 0 c λ xc µ = a λ (b λ xa µ )b µ = λ > µ a λ xb µ = 0 a µ xb λ = 0 µ = (,, ) a µ = a µ b λ = b λ 4.9. λ T, T T T (i, j) a T b T = 0 c T c T = 0 V T V T = 0 T = gt g a T gb T g 0 x A a T xb T = 0 CS d = V dim V λ λ V λ λ (λ ) 2 = d! = S d λ - [5] V λ λ dim V λ V λ = CGc λ S d 2. λ µ V λ V µ S d Proof.. c λ Ac λ = c λ V λ C(c λ ) W V λ c λ W C(c λ ) 0 c λ W = C(c λ ) W AW W V λ = Ac λ = A(c λ W ) = (Ac λ )W W. 73

74 V λ = W c λ W = 0 W W V λ W Ac λ W = 0 Section A W ϕ = ϕ 2 ϕ W Aϕ = W ϕ ϕ W ϕ = ϕ 2 W W = 0 ϕ 0 W = 0 V λ c λ 0 V λ = Ac λ 0 W = V λ V λ 0 c λ V λ = C(c λ ) 2. λ > µ c λ V λ = C(c λ ) c λ V µ = c λ Ac µ = 0 V λ V µ CS d CS d Φ : V λ V µ c λ x 0 x V λ 0 Φ(c λ x) = c λ Φ(x) c λ V µ = λ c λ c λ = d! dim V λ c λ Proof. R c c λ R c : A V λ R c V λ n λ id V λ tr (R c ) = n λ dim V λ c λ = ±g e g S d R c g = gc λ g tr(r c ) = S d = d! n λ dim V λ = d! S d S d S d λ λ V λ λ λ U V λ = V λ U Proof. λ a = a λ, b = b λ, c = c λ V λ = Ac = Aab R a : Aab x xa Aba, R b : Aba y yb Aba 74

75 x Aab x = zab R b R a x = zabab = zc 2 = n λ zc = n λ x c = ba (c ) 2 = n λ c R a R b = n λ id Aab = Aba S d A a(g)g a(g) sgn(g)g A a(g)gba = a(g) sgn(q)gqp g S d g S d,p P,q Q a(g) sgn(p) sgn(g)qp = a(g) sgn(g)g q sgn(p)p q Q q p P sgn(p)p λ λ a λ, b λ V λ = Aa λ b λ Aba = Aa λ b λ Aba S d sgn(g)g S d V λ U = Vλ 4.2 S d S d C d g(z, z 2,, z d ) = (z g (),, z g (d)) C d e i g(e i ) = e g(i) U = C(e + + e d ) U V = { z i e i C d z + + z d = 0}. d v l = e l e l l = 2,, d a 2 v a d v d = a 2 (e 2 e ) + a 3 (e 3 e 2 ) + + a d (e d e d ), = a 2 e + (a 2 a 3 )e 2 + (a 3 a 4 )e (a d a d )e d + a d e d a 2 + (a 2 a 3 ) + (a 2 a 4 ) + + (a d a d ) + a d = 0 V = C{v 2,, v d } 75

76 4.3. V S d k Λ k V 0 k d Proof. C d = V U Λ k (C d ) = (Λ k (U) Λ 0 U) Λ k V Λ U = Λ k V Λ k V χ (χ, χ) = 2 Λ k V C d g S d g k = { 0 g(k) k g(k) = k tr g = d k= g k C d χ (χ, χ) = d! = d! k,l g S d ( g(k)=k and g(l)=l g k )( k l g l ) = d! ( + ) k g(k)=k k l g(k)=k and g(l)=l = ( ) d d! (d {(d )!} + 2 (d 2)!) = 2 2 V Λ k C d A = {,, d} B A #B = k k Λ k V k g S d 0 if g(b) B g B = if g(b) = B g B if g(b) = B g B B = {i,, i k } g(b) = B g(e i e i2 e ik ) = e g(i ) e g(ik ) = g B (e i e ik ) e B = e i e ik {e B B A, #B = k} Λ k C d 76

77 χ = χ Λ k C d(g) = tr (g) = B g B ( ) (χ, χ) = 2 g B d! g G B = g B g C d! = d! g G B,C B,C g(b)=b,g(c)=c sgn(g B ) sgn(g C ) g(b) = B g(c) = C () B C (2) B \ B C (3) C \ B C A \ B C #B C = l (χ, χ) = ( sgna) 2 sgnb sgnc d! B,C a S l b S k l c S k l h S d 2k+l = l!(d 2k + l)! sgnb sgnc d! B C b S k l c S k l b S k l sgnb S k l k l 0, k l = 0, k = l B = C k!(d k)! = ( ) d k!(d k)! = d! d! k B B C l = k ( )( ) d k k (k )!(d k )! = ( ) d k!(d k)! = d! k d! k (χ, χ) = 2 V 4.4. d (d, ) V (d,) S d V Proof. d = (d ) + P = {g S d g(d) = d} = S d, Q = {g S d {g(), g(d)} = {, d}} = S 2 a λ = g(d)=d g, b λ = e (, d) 77

78 c λ = a λ b λ = g(d)=d g h()=d e j = g h, j =, 2,, d g(d)=j h()=j c λ = e d g (j) = i g e j = e i g e j = g g g h = g h = e i g(d)=j h()=j g (d)=i h h ()=i Ac λ = Ae d = span C {e,, e d } g(e + + e d ) = e + + e d g S d e + + e d = 0 span C {e, e 2,, e d } = span C {e 2 e,, e d e d } 4.5. λ = (d s,,, ) V λ Λ s (V ) section [d,, d, d,, d,,,, }{{}}{{}}{{} i d i d i ] i = (i,, i d ) C i C d g S d { 0 g(k) k g k = g(k) = k g C i g(k) = k k i tr C dg = i C = V U χ U (g) = tr V (C i ) = i Λ k C d g S d B {,, d} #B = k 0 if g(b) B g B = if g(b) = B g B if g(b) = B g B 78

79 χ = χ Λ k C d(g) = tr (g) = B g B g g(b) = B B g C i Λ 2 C d ()(2) (i ) i 2 2 = +, 2 = 2 ( ) i tr (g) = i 2 2 tr (g) = tr Λ 2 (V )(g) + tr V (g) χ Λ 2 V (g) = ( ) i i 2 (i ) = 2 2 (i )(i 2) i 2 = ( ) i i 2 2 Λ 3 C d 3 = 3, 3 = 2 +, 3 = + + ( ) i tr (g) = i i 2 + i 3 3 ( ) i χ Λ 3 (V )(g) = i i 2 + i 3 { 3 2 (i )(i 2) i 2 } = ( i 3 ) (i )i 2 + i 3 Λ 4 C d 4 = 4, 4 = 3 +, 4 = 2 + 2, 4 = 2 + +, 4 = ( ) ( ) ( ) i i i2 tr (g) = i 2 + i 3 i + i ( ) ( i i χ Λ 4 (V )(g) = 4 2 ) i 2 + ( ) i2 + (i )i 3 i 4 2 Λ k V k. c.f. Section g = ()(2) (d 2)(d, d) S d i = (d 2, ) = [2,,, ] Λ }{{} k (C d ) d 2 k = + +, k = 2 + } + {{ + } k 2 ( ) ( ) d 2 d 2 χ Λ k C d(g) = k k 2 79

80 k = d 2 k = 0 χ Λ k V (g) =χ Λ k (C d ) χ Λ k (V ) =χ Λ k (C d ) χ Λ k (C d ) + χ Λ k 2 (C d ) + + ( ) k χ V + ( ) k χ U k ( ) ( ) ( ) ( ) d 2 d 2 d 2 d 2 = ( ) s ( ) s = k s k s 2 k k s=0 = d 2k ( ) d d k g = () (d 2)(d, d) χ Λ k (V )(g) = χ Λ d k (V )(g) dim V = d d 2k 0 χ Λ k V (g) 0 Λ k (V ) = Λ d k (V ) 4.3 Frobenius formula 4.3. Frobenius forumla λ V λ S d i = (i, i 2,, i d ), ki k = d, i k 0 d } + {{ + d } + + } + {{ + } i d C i λ l(λ) x,, x k p j (x) j d k i p j (x) = x j + x j x j k (x) (x) = i<j(x i x j ) 80

81 f(x) = f(x,, x k ) [f(x)] l=(l,,l k ) = x l x l k k λ = λ + + λ k λ λ k 0 l = λ + k, l 2 = λ 2 + k 2,, l k = λ k l > l 2 > > l k (Frobenius ). λ V λ χ λ C i [ χ λ (C i ) = (x) ] p j (x) i j (x) p j (x) i j j d (l,,l k ) k(k )/2 + d ki k = k(k )/2 + d k= x l x l k k l i = d + (k )k/2 λ λ k 0 λ l = = λ k = 0 k = l(λ) 0 = λ k+ = λ k+2 = = λ k+s Frobenius l k+ = s,, l k+s = 0 x l k+ k+ xl k+s k+s = xs k+ x0 k+s (x,, x k+s ) (x,, x k+s ) p j (x) i j =(x x 2 ) (x x k )(x x k+ )(x x k+s ) (x 2 x 3 ) (x 2 x k )(x 2 x k+ ) (x 2 x k+s ) (x k x k+ ) (x k x k+s ) (x k+ x k+2 ) (x k+ x k+s ) (x k+s 2 x k+s )(x k+s 2 x k+s ) (x k+s x k+s ) (x + x k + x k+ + x k+s ) i (x d + x d k + x d k+ + x d k+s) i d 8

82 x λ +k+s x λ 2+k+s 2 2 x λ k+s k x s k+ x0 k+s (x) x s x s 2 x s k xs k+ x0 k+s (x x 2 )(x x 3 ) (x x k ) (x 2 x 3 ) (x 2 x k ) (x k x k ) (x + x k + x k+ + x k+s ) i (x d + x d k + x d k+ + x d k+s) i d x λ +k x λ 2+k 2 2 x λ k k x,, x k 4.9. Section 5 S λ P i = P (i,i 2,,i d ) = χ λ (C i )S λ j d p j (x) ij = λ λ d Frobenius Section 6 Section 6 Section S 3 C (3,0,0), C (,,0), C (0,0,) C (3,0,0), ()(2, 3) C (,,0), (, 2, 3) C (0,0,) x, x 2, x 3 p j (x) = x j + x j 2 + x j 3 (x) = (x x 2 )(x 2 x 3 )(x x 3 ). C (3,0,0) (x) j p j(x) i j f(x) = (x)(x + x 2 + x 3 ) 3 (x 2 + x x 2 3) 0 (x 3 + x x 3 3) 0 χ (3) (C (3,0,0) ) = [f(x)] (3+3,0+3 2,0+3 3) = [f(x)] (5,,0) = χ (,,) (C (3,0,0) ) = [f(x)] (+3,+3 2,+3 3) = [f(x)] (3,2,) = χ (2,) (C (3,0,0) ) = [f(x)] (4,2,0) = 2 idc (3,0,0) 82

83 2. C (,,0) f(x) = (x)(x + x 2 + x 3 )(x 2 + x x 2 3) χ (3) (C (,,0) ) = [f(x)] (5,,0) = χ (,,) (C (,,0) ) = [f(x)] (3,2,) = χ (2,) (C (,,0) ) = [f(x)] (4,2,0) = 0 3. C (0,0,) f(x) = (x)(x 3 + x x 3 3) χ (3) (C (0,0,) ) = [f(x)] (5,,0) = χ (,,) (C (0,0,) ) = [f(x)] (3,2,) = χ (2,) (C (0,0,) ) = [f(x)] (4,2,0) = see Section S 3 σ = (, 2) τ = (, 2, 3) U U V 2 0 = ω + ω S d V (d,) l(λ) 2 x, x 2 l = d + 2 = d, l 2 = = χ (d,) (C (i,i 2,,i d )) = [(x x 2 )(x + x 2 ) i (x 2 + x 2 2) i2 (x d + x d 2) i d ] (d,) = [(x x 2 )(x i + i x i x x i 2 )(x 2 + x 2 2) i2 (x d + x d 2) i d ] (d,) = i Section χ λ (e) e C (d,0,,0) dim V λ = χ λ (C (d,0,,0) ) = [ (x)(x + + x k ) d ] (l,,l k ) 83

84 k [ (x)p (x) d ] l (x) x k x k k (x) =... x x k = sgn(σ)x σ() k x σ(k) σ S k Proof. (x) (x i x j ) x k x k τ S k sgn(τ)a τ() a kτ(k) = (x + + x k ) d = τ S k r + +r k =d sgn(τ)x τ() k x τ(k) d! r! r k! xr x r k k (x)(x + + x k ) d = sgn(σ)x σ() k x σ(k) σ S k d! = sgn(σ) r! r k! xr +σ(k) x r i+σ(k i+) i σ S k,r + +r k =d r + +r k =d d! r! r k! xr x r k k x r k+σ() k x,, x k d + k(k ) x l 2 x l k k d + k(k ) 2 sgn(σ) d! (l σ(k) + )! (l k σ() + )! σ S k l k i+ σ(i) + 0 i k or l j σ(k j + ) + 0 j k d! l! l k! = d! l! l k! σ S k sgn(σ) k l j (l j ) (l j σ(k j + ) + 2) j= l k l k (l k ).... = l l (l ) d! l! l k! (l i l j ) i<j 84

85 Proof. d! sgn(σ) (l σ(k) + )! (l k σ() + )! = d! sgn(σ)l (l ) (l σ(k) + 2) l k (l k ) (l k σ() + 2) l! l k! = d! sgn(σ)l (l ) (l σ(k) + 2) l k (l k ) (l k σ() + 2) l! l k! σ S k σ S k l k i+ σ(i) + 0 i k σ S k l k i+ σ(i) + = 0 l k l k (l k ) l k (l k ) (l k k + 2)..... = sgn(σ)a σ() a kσ(k) l l (l ) l (l ) (l k + 2) = sgn(σ){l k (l k ) (l k σ() + 2)} {l (l ) (l σ(k) + 2)} l i = l j (l i l j ) i<j (l i l j ) k(k )/2 k(k )/2 det = c i<j (l i l j ) l k l2 k 2 l k c = 4.22 (). λ = λ + + λ k S d V λ d! dim V λ = (l i l j ), l i = λ i + k i l! l k! i<j hook length forumla hook hook hook length hook length 85

86 (hook length formula). dim V λ = dim V λ = each box d! (hook lengths) 8! = 70 Proof. λ + + λ k l(λ) = k hook length l = λ + k, l 2 = λ 2 + k 2,, l k = λ k l(λ) = d λ = + + V λ = = d! d!(d )!! d! d!(d )!! i<j d (j i) = i<j ( + d i ( + d j)) hook length hook length λ = + + p d! l! l k! (l i l j ) = i<j each box d! (hook lengths) λ = λ + + λ k λ p d k λ 86

87 λ l(λ ) k p (d k)! (l )! (l k )! dim V λ = d! l! l k! (l i l j ) = i<j (l i l j ) = i<j d! (d k)! l l 2 l k = d! (d k)! (d k)! l l 2 l k d! = (hook lengths) each box in λ each box in λ (hook lengths) (d k)! each box in λ (hook lengths) (d k)! (l )! (l k )! (l i l j ) i<j Section i = (d) χ λ (C i ) = χ λ (e) = dim V λ i = (d m, 0,, 0, }{{}, 0,, 0) m χ λ (C i ) g = (,, m) S d χ λ (g) [ (x)(x + + x k ) d m (x m + + x m k )] l (x)(x + + x k ) d m i x m i = i σ S k,r + +r k =d m (d m)! sgn(σ) r! r k! xr +σ(k) x r i+σ(k i+) +m i x r k+σ() k x l x l k k (d m)! sgn(σ) (l σ(k) + )! (l i σ(k i + ) + m)! (l k σ() + )! i S i k 87

88 S i k σ S k l j σ(k j + ) + 0 j i l i σ(k i + ) + m 0 { } (d m)! sgn(σ) l j (l j ) (l j σ(k j + ) + 2) l i! l k! S k j i = i {l i (l i m)(l i m)(l i 2 m) (l i σ(k i + ) + 2 m)} (d m)! l! l k! {l i (l i m + )} S k sgn(σ){ j i (l i m) (l i σ(k i + ) + 2 m) l j (l j ) (l j σ(k j + ) + 2)} S k l k l k (l k ) l k (l k k + 2)..... l i m (l i m)(l i m ) (l i m) (l i k + 2 m)..... l l (l ) l (l k + 2) s i, t i l s = l t (l s l t ) s i l s = l i m (l i l s m) s<t c (l s l t ) k s= (l i l s m) m s i (l, c i l s ) l k lk 2 2 lk l k lk 2 2 lk c = = χ λ (g) = m 2 d! (d m)!m = dim V λ m 2 h m (d m)! m l! l k! k i= h m = d! s<t (l s l t ) l! l k! ψ(l i ) ϕ (l i ) ϕ(x) = s= i m (l i s + ) s= s<t (l s l t ) k s= (l i l s m) s i (l i l s ) k s= (l i l s m) m s= (l i s + ) s i (l i l s ) i k m (x l s ), ψ(x) = ϕ(x m) (x s + ) d! (d m)!m g = (,, m) S d m > 88 s=

89 4.24. g = (,, m) χ λ (g) = dim V λ m 2 h m k i= ψ(l i ) ϕ (l i ). k i= ψ(l i)/ϕ (l i ) ψ(x)/ϕ(x) x = c n /x n x Proof. r Res( ) = f(z)dz = c 2πi z =r f(z)dz = 2πi z =r i Res(a i ) ψ(z)/ϕ(z) z = l i Res(l i ) = (z l i ) ψ(z) k s= ϕ(z) = (l i l s m) m s= (l i s + ) z=li s i (l i l s ) k i= ψ(l i)/ϕ (l i ) ψ(z)/ϕ(z) z z = z = a a f(z) k f(z) = c k (z a) + + c k z a + a n (z a) n Res(a, f) = c c = d k f(z)dz = 2πi (k )! dz (z k a)k f(z) z=a z a =r n=0 f(z)dz = C i Res(a i, f) C a,, a s z = C { } = P (C) f(z)dz f(z)dz = f(/w)d(/w) = f(/w)w 2 dw 89

90 f(z) z = f(z) = c k z k + + c z + c n /z n f(/w)w 2 = c k w k+2 c /w 3 c 0 /w 2 c /w c n /z n c P (C) Res(ai, fdz) = λ λ rank b,, b r a,, a r a < a 2 < < a r, b < b 2 < < b r rank= 4 b 4 = 9 b 3 = 7 b 2 = 6 b = 0 a 4 = 6 a 3 = 4 a 2 = 3 a = 2 ( ) a a 2 a r b b 2 b r λ characteristics ( ) λ {l,, l k, k a,, k a r } = {0,,, k, k + b,, k + b r } l i = λ i + k i 90

91 Proof. rank r l = k + b r, l 2 = k + b r,, l r = k + b {l r+,, l k, k a,, k a r } = {0,,, k } k a r, k a r,, k a l k, l k+,, l r l i = λ i + k i k a i f(y) = r (y b i ) m i=, g(y) = f(y m) (y j + ) r j= (y + a i + ) i= ψ(x)/ϕ(x) = g(y)/f(y), y = x k k i= ψ(l i)/ϕ (l i ) g(y)/f(y) y = y y = y = x k Proof. k r (x l i ) (x k + + a i ) = i= i= k r (x i + ) (x k b i ) i= i= 9

92 ϕ(x) = k (x l i ) = i= (x i + ) (x k bi ) (x k + + ai ) = k (x i + )f(x k) i= = ψ(x)/ϕ(x) = ϕ(x m) m (x j + )/ϕ(x) j= k i= (x m i + )f(x m k) m f(y m) = f(y) f(y m) = f(y) k i= j= (x i + )f(x k) k m (x m i + ) (x i + ) i= m l=m k+ i=k+ (x j + ) m k (x k l + ) (x k l + ) = g(y) f(y) l= g = (, 2,, m) λ χ λ f(y), g(y) λ m χ λ (g) = dim V m 2 h m ( Res(, g(y)/f(y))) Res(, g(y)/f(y)) y = g(y)/f(y) g(y)/f(y) = c n /y n y = Res(, g(y)/f(y)) = c m = 2 g = (, 2) χ λ (g) = dim V λ d(d ) r (b i (b i + ) a i (a i + )) i= Proof. χ λ (g) = dim V Res(, g(y)/f(y)) 2d(d ) 92

93 Res(, g(y)/f(y)) y = y = /w g(y) (y 2 f(y) = bi ) (y + ai + ) y(y ) (y + ai ) (y bi ) ( 2w bi w) ( + ai w + w) = ( w)(/w) 2 ( + ai w w) ( bi w) w ( d 3 ( 2w bi w) ( + ai w + w) ( w) 3! dw 3 ( + ai w w) ( bi w) = 6 {( ( 2w bi w) ( + ai w w) 3 ( ( 2w bi w) ( + ai w w) ) ( + ai w + w) w=0 ( bi w) ) ( + ai w + w) w=0} ( bi w) ) w=0 H (w) = ( 2w b i w) (+a i w+w), H 2 (w) = (+a i w w) ( b i w) A = r + (a i b i ) H (0) = H (0) =, H 2 (0) = r r ( 2 b i ) + a i + = r + (a i b i ) = A i= i= H 2(0) = a i b i = r + (a i b i ) = A H (0) = i ( 2 b i )(2 + b i + j (a j b j )) + (a i + )( a i + j (a j b i )), =A 2 i (2 + b i ) 2 + (a i + ) 2 H 2 (0) = i (a i )( a i + + j (a j b j )) b i (b i + j (a j b j )) =A 2 i (a i ) 2 + b 2 i 93

94 H (0) = (b i + 2)(b j + 2)(4 + b i + b j + A) i j i,j 2(b i + 2)(a j + )(b i a j + + A) + i j (a i + )(a j + )( a i a j 2 + A) = i (b i + 2) 2 (4 + 2b i + A) i (a i + ) 2 ( 2a i 2 + A) + i,j {(b i + 2)(b j + 2)(4 + b i + b j + A) 2(b i + 2)(a j + )(b i a j + + A) + (a i + )(a j + )( a i a j 2 + A)} H 2 (0) = b i b j (b i + b j + A) 2 i j i,j b i (a j )(b i a j + + A) + i j (a i )(a j )( a i a j A) = i b 2 i (2b i + A) i (a i ) 2 ( 2a i A) + i,j {b i b j (b i + b j + A) 2b i (a j )(b i a j + + A) + (a i )(a j )( a i a j A)} (H (w)h 2 (w) ) w=0 =H (0)H 2 (0) + 3H (0)(H 2 (0) ) + 3H (0)(H 2 (0) ) + (H 2 (0) ) H (0) =H (0) 3H (0)H 2(0) + 3H (0)(2H 2(0) 2 H 2 (0)) H 2 (0) + 6H 2(0)H 2 (0) 6H 2(0) 3 =H (0) H 2 (0) + 2A + b i + a i =8 i,j ( a i a j + 2b i + b i b j ) (b i + 2) 2 (4 + 2b i + A) (a i + ) 2 ( 2a i 2 + A) + b 2 i (2b i + A) + (a i ) 2 ( 2a i A) + 2A + b i + a i =8 i,j ( a i a j + 2b i + b i b j ) + 4 ( + a i + b i )(3a i 3b i + 2A 3) 3(H (w)h 2 (w) ) w=0 = 3(H (0) 2H (0)H 2(0) + 2(H 2(0)) 2 H 2 (0)) = 3(A 2 i (2 + b i ) 2 + (a i + ) 2 A 2 + i (a i ) 2 + b 2 i ) = 2 + b i + a i 94

95 8 i,j =8 i,j ( a i a j + 2b i + b i b j ) + 4 ( + a i + b i )(3a i 3b i + 2A 3) b i + a i ( a i a j + 2b i + b i b j ) + 4 ( + a i + b i )(3a i 3b i + 2A) =2 ( + a i + b i )(a i b i ) = 2 (b i (b i + ) a i (a i + )) χ λ (g) = dim V (bi (b i + ) a i (a i + )) d(d ) g = (, 2,, d) S d λ χ λ (g) = ± λ χ λ (g) = 0 χ λ (g) = { ( ) s if λ = (d s,,, ), 0 s d 0 otherwise Proof. Frobenius d Frobenius (x)(x d + + x d d ) xl x l 2 2 x l d d l i = λ i + d i l > l 2 > l d 0 (x) = sgn(σ)x σ() d x σ(d) (x)(x d + x d d) = d p= σ sgn(σ)x σ() d x σ(d) x d p x l x l 2 2 x l d d l > l 2 > l d 0 sgn(σ)x σ() d x σ(d) x d σ ( ) d d d 2 t t σ = t d d t + t 95