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

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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