λ 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

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1 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 > µ k Young Young semi-standard Young λ Young Young STab(λ), SSTab(λ) λ/µ skew Young skew Young STab(λ/µ), SSTab(λ/µ) λ numbering λ Young λ tshun@kurims.kyoto-u.ac.jp

2 λ 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 SSTab(µ) cont T = λ} λ m, µ n, ν m + n Littlewood-Richardson c ν λµ 2 c ν λµ def = #{T SSTab(ν/λ) cont T = µ word T Yamanouchi } 0.3. Littlewood-Richardson m = n = 3, λ = µ = (2, ) = ν 6 Littlewood-Richardson c ν λµ c (5,) (2,)(2,) = 0 c(4,2) (2,)(2,) = c(4,,) (2,)(2,) = c(3,3) (2,)(2,) = c(3,,,) (2,)(2,) = c (3,2,) (2,)(2,) = 2 c(2,2,2) (2,)(2,) = c(2,2,,) (2,)(2,) = c(2,,,,) (2,)(2,) = 0 C- C[S n ] C[S n ] Young.. Young 2 Yamanouchi reverse lattice word w Yamanouchi k l d k l k i < j =) d k i dk j 2

3 λ = (λ,, λ l ) n T λ STab(λ) λ T λ = λ + λ + λ 2. n λ l + n S n λ n numbering Num(λ) (σ T) ij σ S n, T Num(λ) def = σ(t ij ) λ n numbering T Num(λ) R(T) C(T) R(T) = {σ S n σ T T } C(T) = {σ S n σ T T } a T b T Young c T C[S n ] a T = σ, b T = (sgn σ)σ, σ R(T) σ C(T ) c T = a T b T λ n a λ = a Tλ, b λ = b Tλ, c λ = c Tλ, R λ = R(T λ ), C λ = C(T λ ) λ n C[S n ] S λ S λ = C[S n ]c λ λ = (λ,, λ l ) S λ def = S λ S λl S λ S n M λ = Ind Sn S λ C C T Num(λ) a T a λ.2. S, T λ, µ n numbering λ µ i, j 2 T S λ = µ p R(S), q C(S) T = p q S T S λ µ λ = µ q C(S) q S T q C(S) q S T p R(S) T = p q S 3

4 .3.Young λ n c C[S n ] p R λ, q C λ p c (sgn q)q = c β c = βc λ β c = βc λ R λ C λ = {} σ S n p R λ, q C λ σ = pq c = σ S n γ σ σ p R λ, q C λ p c (sgn q)q = c p R λ, q C λ, σ S n γ pσq = (sgn q)γ σ γ pq = (sgn q)γ σ S n p R λ, q C λ σ = pq γ σ = 0.2. i, j 2 σ T λ T λ p 0 = σ(i, j)σ R λ, q 0 = (i, j) C λ γ σ = γ p0 σq 0 = (sgn q 0 )γ σ = γ σ σ R λ C λ γ σ = 0.4. Young λ, µ n λ > µ c λ c µ = 0 x C[S n ] a λ xb µ = 0 σ S n a λ σb µ = 0 a λ σb µ = a λ σb µ σ σ = a Tλ b σ Tµ σ λ µ λ µ λ µ..2. i, j 2 T λ σ T µ π = (i, j) a Tλ b σ Tµ = (a Tλ π) (πb σ Tµ ) = (a Tλ ) ( b σ Tµ ) = a Tλ b σ Tµ.5. S n {S λ λ n}.3. β λ C c λ c λ = β λ c λ λ n β λ 0 c λ 0 C dim Hom C[Sn ](S λ, S λ ) = dim c λ C[S n ]c λ = dim Cc λ = S λ λ µ λ > µ.4. dim Hom C[Sn ](S λ, S µ ) = dim c λ C[S n ]c µ = dim{0} = 0 4

5 C[S n ]- S λ = S µ S n S n λ n β λ 0 ρ : C[S n ] C[S n ], x xc λ C[S n ] A = S λ I C[S n ]- I Im ρ S λ Trace ρ = β λ dim S λ R λ C λ = {} Trace(ρ) = (sgn q) Trace(pq) = G p R λ,q C λ G = β λ dim S λ β λ 0 S λ Specht Specht n C[x,, x n ] S n λ n S n {M λ λ n} M λ S n /R λ M λ = C[S n ]a λ S λ polytabloid dim S λ = #STab(λ) {M λ λ n} {S λ λ n} λ P + λ Verma M(λ) λ V(λ) Kac-Weyl 2.6. Young k µλ M λ M λ = S λ (S µ ) k µλ µ n,µ>λ M λ S λ a λ a λ = #R λ a λ dim Hom C[Sn ](S λ, M λ ) = dim Hom C[Sn ](M λ, S λ ) = dim a λ C[S n ]c λ = dim Cc λ = 2.3. µ < λ dim Hom C[Sn ](S µ, M λ ) = dim a µ b µ C[S n ]a λ = dim{0} = 0 S µ M λ 2 µ λ.2. b µ C[S n ]a λ = 0 k µλ Kostka K µλ 5

6 .7. M λ λ = (λ,, λ n ), µ = ( m,, n mn ) n M λ C(µ) n (x i + + x i n) m i x λ xλn n i= M λ tabloid Young 2 S, S 2, S n Grothendieck Z- 2.. S n V [V] Z- S n Grothendieck R n {[V] V S n } [V W] = [V] + [W] R = n 0 R n R 0 = ZS m VS n W [V] [W] def = [Ind Sm+n S m S n V C W]( R m+n ) Z-S 0 = {} R 0 [] S 0 3 R λ, µ P [S λ ], [S µ ] = δ λµ Z m n ϕ m,n : Z[x,, x m ] Z[x,, x n ], x i ( i n) x i 0 (n < i m) n Z[x,, x n ] S n Λ n k 0 Λ k n 3 6

7 ϕ m,n Λ k m ρk def m,n = ϕ Λ m,n k : Λ k m Λ k n m k 0 {ρ k m,n : Λ k m Λ k n m n} Λ k def = lim Λ k k n, Λ def = Λ k k 0 Λ Z Schur λ = (λ,, λ l ) P m λ Λ λ m λ = i,,i l x λ i x λ l i l (λ,, λ l ) n n p n Λ n h n Λ n p n = k x n k, h n = λ n m λ λ = (λ,, λ l ) P p λ = p λ p λl ( Λ λ ), h λ = h λ h λl ( Λ λ ) λ = (λ,, λ l ) P Schur m λ Λ λ s λ = T SSTab(λ) x T ( x T def = x #{(i,j) T ) ij=k} k k [M] Λ k, Λ Λ k = λ k Zm λ, Λ = λ P Zm λ Λ Λ k, Λ Λ 4 the ring of symmetric functions 5 Schur Schur s (x ; ; x m ) = det((x j ) i+m-i ) i;j m = det((x j ) m-i ) i;j m Weyl 7

8 Λ (x + x 2 ) 2 = (x 2 + x 2 2) + 2(x x 2 ) (x + x 2 + x 3 ) 2 = (x 2 + x x 2 3) + 2(x x 2 + x x 3 + x 2 x 3 ) etc Λ Λ m 2 = m + 2m ^Λ ρ m,n def = ϕ m,n Λm : Λ m Λ n {ρ m,n : Λ m Λ n m n} ^Λ = lim n Λ n 2.2. Λ ^Λ ^Λ = λ P Zm λ Λ Λ k A A Λ Λ A Λ A = Λ Z A 2.5. λ n h λ = s λ + K µλ s µ µ n,µ λ Robinson-Schensted-Knuth RSK Young 2.5. M λ 2.6. Λ {h λ λ P}, {s λ λ P} Λ Z- {p λ λ P} Λ Q = Λ Q Q- c µλ s λ = m λ + µ<λ c µλ m µ {m λ λ P} Λ Z- {s λ λ P} Λ 2.5. {h λ λ P} Λ {p λ λ P} nh n p h n p 2 h n 2 p n = 0 8

9 Newton n 0 h n t n = i x i t, n 0 p n tn n = log i x i t Λ λ, µ P s λ, s µ = δ λµ Z λ, µ P h λ, m µ = δ λµ, p λ, p µ = z λ δ λµ λ = ( m,, k m k ) P z λ def = k j mj m j! 6 i,j x i y j = λ P h λ (x)m λ (y) = λ P j= z λ p λ (x)p λ (y) RSK Young i,j x i y j = λ P s λ (x)s λ (y) Cauchy-Littlewood (GL, GL) λ, µ n χ λ µ, ξ λ µ Z p µ = λ n χ λ µs λ = λ n ξ λ µm λ λ n s λ = µ n z µ χ λ µp µ, h λ = µ n z µ ξ λ µp µ 2.. Ψ : Λ R, s λ [S λ ] Z- Z-, 6 n #C( ) = n!=z 9

10 [M] [F] ^Ψ : Λ R, h λ [M λ ] Z- Z-, ^Ψ(s λ ) = [S λ ] ^Ψ = Ψ ^Ψ Z- λ = (λ,, λ l ) h λ = h λ h λl 2.6. [M (λ ) ] [M (λ l) ] = [M λ ] [M (λ ) ] [M (λ 2) ] = [C[S λ +λ 2 ] C[Sλ S λ2 ] C C C] = [C[S λ +λ 2 ] C[S (λ,λ 2 ) ] C] = [Ind S λ +λ 2 S (λ,λ 2 ) C] = [M (λ,λ 2 ) ] ^Ψ.6. {[M λ ] λ P} R Z ^Ψ ^Ψ ^Ψ ^Ψ, ^Ψ V S n χ C(µ) χ(c(µ)) 2.0. [M λ ] h λ = ξ λ z µp µ µ µ.7. ξ λ µ Mλ C(µ) Θ : R Λ Q, [V] µ n z µ χ(c(µ))p µ Θ Ψ : Λ Λ Q () Ψ R Θ : R Λ Θ, V, W S n f, g V, W Θ([V]), Θ([W]) = λ,µ n z λ z µ f(c(λ)) g(c(µ)) p λ, p µ 2.8. = f(c(λ)) g(c(λ)) z λ λ n = f(σ) g(σ ) = [V], [W] n! σ S n Θ ^Ψ, 0

11 ^Ψ(s λ ) = [S λ ] h λ = s λ + µ λ K µλ s µ, M λ = S λ µ>λ(s µ ) k µλ m µλ ^Ψ(s λ ) = [S λ ] + µ>λ m µλ [S µ ] ^Ψ, = s λ, s λ = ^Ψ(s λ ), ^Ψ(s λ ) = + µ>λ m 2 µλ ^Ψ(s λ ) = [S λ ] V S n χ V Θ : R Λ, [V] µ n z µ χ(c(µ))p µ ch = Θ (characteristic map) Young 2.5. λ n M λ = S λ µ n,µ λ (S µ ) K µλ Frobenius λ, µ n S n S λ C(µ)- χ λ µ S λ Gramer Murnaghan-Nakayama 2.5. Littlewood-Richardson λ m, µ n Ind S m+n S m S n S λ C S µ = ν m+n (S ν ) cν λµ

12 2.5. λ m, µ n Schur s λ s µ = ν m+n c ν λµs ν Young RSK Schützenberger jeu de taquin 7, Knuth 2.6. Pieri 2.5. µ = () λ n Ind S n+ S n S λ = λ µ µ Young λ Young λ µ S µ [F] William Fulton. Young Tableaux. Cambridge University Press, 997 [M] I. G. Macdonald. Symmetric Functions and Hall Polynomials. Oxford University Press, 995 [W] Hermann Weyl. The Classical Groups. Princeton University Press,

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