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. Drinfeld type realization of cyclotomic q -Schur algebras . Kentaro Wada Shinshu Univ. 12th March, 2012 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 1 / 25

slide-1
SLIDE 1

. . . . . .

. .

Drinfeld type realization of cyclotomic q-Schur algebras

Kentaro Wada

Shinshu Univ.

12th March, 2012

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 1 / 25

slide-2
SLIDE 2

. . . . . .

Introduction

Hn,r : Ariki-Koike alg. /a field ass. to Sn ⋉ (Z/rZ)n. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r. Sn,r : a quasi-hereditary cover of Hn,r.

Schur-Weyl duality (r = 1)

Uq(glm) ↷ V⊗n ↶ Hn,1

This duality holds for any field amd parameter.

Im (Uq(glm) → End(V⊗n)) EndHn,1(V⊗n) Sn,1.

Schur-Weyl duality (r > 1)

Uq(glm) ↷V⊗n ↶ Hn,1 ∪ ∩ Uq(g) ↷V⊗n ↶ Hn,r (g = glm1 ⊕ · · · ⊕ glmr ⊂ glm )

This duality holds only the case Hn,r : semi-simple.

“ V⊗n ” : too small !

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 2 / 25

slide-3
SLIDE 3

. . . . . .

Introduction

Hn,r : Ariki-Koike alg. /a field ass. to Sn ⋉ (Z/rZ)n. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r. Sn,r : a quasi-hereditary cover of Hn,r.

Schur-Weyl duality (r = 1)

Uq(glm) ↷ V⊗n ↶ Hn,1

This duality holds for any field amd parameter.

Im (Uq(glm) → End(V⊗n)) EndHn,1(V⊗n) Sn,1.

Schur-Weyl duality (r > 1)

Uq(glm) ↷V⊗n ↶ Hn,1 ∪ ∩ Uq(g) ↷V⊗n ↶ Hn,r (g = glm1 ⊕ · · · ⊕ glmr ⊂ glm )

This duality holds only the case Hn,r : semi-simple.

“ V⊗n ” : too small !

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 2 / 25

slide-4
SLIDE 4

. . . . . .

Introduction

Hn,r : Ariki-Koike alg. /a field ass. to Sn ⋉ (Z/rZ)n. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r. Sn,r : a quasi-hereditary cover of Hn,r.

Schur-Weyl duality (r = 1)

Uq(glm) ↷ V⊗n ↶ Hn,1

This duality holds for any field amd parameter.

Im (Uq(glm) → End(V⊗n)) EndHn,1(V⊗n) Sn,1.

Schur-Weyl duality (r > 1)

Uq(glm) ↷V⊗n ↶ Hn,1 ∪ ∩ Uq(g) ↷V⊗n ↶ Hn,r (g = glm1 ⊕ · · · ⊕ glmr ⊂ glm )

This duality holds only the case Hn,r : semi-simple.

“ V⊗n ” : too small !

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 2 / 25

slide-5
SLIDE 5

. . . . . .

Introduction

Hn,r : Ariki-Koike alg. /a field ass. to Sn ⋉ (Z/rZ)n. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r. Sn,r : a quasi-hereditary cover of Hn,r.

Schur-Weyl duality (r = 1)

Uq(glm) ↷ V⊗n ↶ Hn,1

This duality holds for any field amd parameter.

Im (Uq(glm) → End(V⊗n)) EndHn,1(V⊗n) Sn,1.

Schur-Weyl duality (r > 1)

Uq(glm) ↷V⊗n ↶ Hn,1 ∪ ∩ Uq(g) ↷V⊗n ↶ Hn,r (g = glm1 ⊕ · · · ⊕ glmr ⊂ glm )

This duality holds only the case Hn,r : semi-simple.

“ V⊗n ” : too small !

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 2 / 25

slide-6
SLIDE 6

. . . . . .

Introduction

Hn,r : Ariki-Koike alg. /a field ass. to Sn ⋉ (Z/rZ)n. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r. Sn,r : a quasi-hereditary cover of Hn,r.

Schur-Weyl duality (r = 1)

Uq(glm) ↷ V⊗n ↶ Hn,1

This duality holds for any field amd parameter.

Im (Uq(glm) → End(V⊗n)) EndHn,1(V⊗n) Sn,1.

Schur-Weyl duality (r > 1)

Uq(glm) ↷V⊗n ↶ Hn,1 ∪ ∩ Uq(g) ↷V⊗n ↶ Hn,r (g = glm1 ⊕ · · · ⊕ glmr ⊂ glm )

This duality holds only the case Hn,r : semi-simple.

“ V⊗n ” : too small !

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 2 / 25

slide-7
SLIDE 7

. . . . . .

Introduction

Hn,r : Ariki-Koike alg. /a field ass. to Sn ⋉ (Z/rZ)n. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r. Sn,r : a quasi-hereditary cover of Hn,r.

Schur-Weyl duality (r = 1)

Uq(glm) ↷ V⊗n ↶ Hn,1

This duality holds for any field amd parameter.

Im (Uq(glm) → End(V⊗n)) EndHn,1(V⊗n) Sn,1.

Schur-Weyl duality (r > 1)

Uq(glm) ↷V⊗n ↶ Hn,1 ∪ ∩ Uq(g) ↷V⊗n ↶ Hn,r (g = glm1 ⊕ · · · ⊕ glmr ⊂ glm )

This duality holds only the case Hn,r : semi-simple.

“ V⊗n ” : too small !

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 2 / 25

slide-8
SLIDE 8

. . . . . .

Introduction

Hn,r : Ariki-Koike alg. /a field ass. to Sn ⋉ (Z/rZ)n. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r. Sn,r : a quasi-hereditary cover of Hn,r.

Schur-Weyl duality (r = 1)

Uq(glm) ↷ V⊗n ↶ Hn,1

This duality holds for any field amd parameter.

Im (Uq(glm) → End(V⊗n)) EndHn,1(V⊗n) Sn,1.

Schur-Weyl duality (r > 1)

Uq(glm) ↷V⊗n ↶ Hn,1 ∪ ∩ Uq(g) ↷V⊗n ↶ Hn,r (g = glm1 ⊕ · · · ⊕ glmr ⊂ glm )

This duality holds only the case Hn,r : semi-simple.

“ V⊗n ” : too small !

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 2 / 25

slide-9
SLIDE 9

. . . . . .

Introduction

r = 1 ( Sn,1 EndHn,1(V⊗n) ) V⊗n = ⊕

µ V⊗n µ

: weight space decom. as Uq(glm)-module.

V⊗n

µ IndHn,1 H (Sµ) 1

as Hn,1-module (permutation module).

r > 1 Mµ := mµ · Hn,r

(mµ ∈ Hn,r) “ permutation module”

Sn,r := EndHn,r ( ⊕

µ Mµ)

; cyclotomic q-Schur alg. Today Introduce an algebra U ass. to Cartan data of glm s.t. U ↠ Sn,r (Drinfeld type presentation). Representation theory of U and Sn,r.

highest weight modules. Harish-Chandra Ind and Res (UL ֒→ UP ֒→ U) evaluation functor Og → UL -mod (Og ⊂ Uq(g) -mod).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 3 / 25

slide-10
SLIDE 10

. . . . . .

Introduction

r = 1 ( Sn,1 EndHn,1(V⊗n) ) V⊗n = ⊕

µ V⊗n µ

: weight space decom. as Uq(glm)-module.

V⊗n

µ IndHn,1 H (Sµ) 1

as Hn,1-module (permutation module).

r > 1 Mµ := mµ · Hn,r

(mµ ∈ Hn,r) “ permutation module”

Sn,r := EndHn,r ( ⊕

µ Mµ)

; cyclotomic q-Schur alg. Today Introduce an algebra U ass. to Cartan data of glm s.t. U ↠ Sn,r (Drinfeld type presentation). Representation theory of U and Sn,r.

highest weight modules. Harish-Chandra Ind and Res (UL ֒→ UP ֒→ U) evaluation functor Og → UL -mod (Og ⊂ Uq(g) -mod).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 3 / 25

slide-11
SLIDE 11

. . . . . .

Introduction

r = 1 ( Sn,1 EndHn,1(V⊗n) ) V⊗n = ⊕

µ V⊗n µ

: weight space decom. as Uq(glm)-module.

V⊗n

µ IndHn,1 H (Sµ) 1

as Hn,1-module (permutation module).

r > 1 Mµ := mµ · Hn,r

(mµ ∈ Hn,r) “ permutation module”

Sn,r := EndHn,r ( ⊕

µ Mµ)

; cyclotomic q-Schur alg. Today Introduce an algebra U ass. to Cartan data of glm s.t. U ↠ Sn,r (Drinfeld type presentation). Representation theory of U and Sn,r.

highest weight modules. Harish-Chandra Ind and Res (UL ֒→ UP ֒→ U) evaluation functor Og → UL -mod (Og ⊂ Uq(g) -mod).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 3 / 25

slide-12
SLIDE 12

. . . . . .

Introduction

r = 1 ( Sn,1 EndHn,1(V⊗n) ) V⊗n = ⊕

µ V⊗n µ

: weight space decom. as Uq(glm)-module.

V⊗n

µ IndHn,1 H (Sµ) 1

as Hn,1-module (permutation module).

r > 1 Mµ := mµ · Hn,r

(mµ ∈ Hn,r) “ permutation module”

Sn,r := EndHn,r ( ⊕

µ Mµ)

; cyclotomic q-Schur alg. Today Introduce an algebra U ass. to Cartan data of glm s.t. U ↠ Sn,r (Drinfeld type presentation). Representation theory of U and Sn,r.

highest weight modules. Harish-Chandra Ind and Res (UL ֒→ UP ֒→ U) evaluation functor Og → UL -mod (Og ⊂ Uq(g) -mod).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 3 / 25

slide-13
SLIDE 13

. . . . . .

Introduction

r = 1 ( Sn,1 EndHn,1(V⊗n) ) V⊗n = ⊕

µ V⊗n µ

: weight space decom. as Uq(glm)-module.

V⊗n

µ IndHn,1 H (Sµ) 1

as Hn,1-module (permutation module).

r > 1 Mµ := mµ · Hn,r

(mµ ∈ Hn,r) “ permutation module”

Sn,r := EndHn,r ( ⊕

µ Mµ)

; cyclotomic q-Schur alg. Today Introduce an algebra U ass. to Cartan data of glm s.t. U ↠ Sn,r (Drinfeld type presentation). Representation theory of U and Sn,r.

highest weight modules. Harish-Chandra Ind and Res (UL ֒→ UP ֒→ U) evaluation functor Og → UL -mod (Og ⊂ Uq(g) -mod).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 3 / 25

slide-14
SLIDE 14

. . . . . .

Introduction

r = 1 ( Sn,1 EndHn,1(V⊗n) ) V⊗n = ⊕

µ V⊗n µ

: weight space decom. as Uq(glm)-module.

V⊗n

µ IndHn,1 H (Sµ) 1

as Hn,1-module (permutation module).

r > 1 Mµ := mµ · Hn,r

(mµ ∈ Hn,r) “ permutation module”

Sn,r := EndHn,r ( ⊕

µ Mµ)

; cyclotomic q-Schur alg. Today Introduce an algebra U ass. to Cartan data of glm s.t. U ↠ Sn,r (Drinfeld type presentation). Representation theory of U and Sn,r.

highest weight modules. Harish-Chandra Ind and Res (UL ֒→ UP ֒→ U) evaluation functor Og → UL -mod (Og ⊂ Uq(g) -mod).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 3 / 25

slide-15
SLIDE 15

. . . . . .

Introduction

r = 1 ( Sn,1 EndHn,1(V⊗n) ) V⊗n = ⊕

µ V⊗n µ

: weight space decom. as Uq(glm)-module.

V⊗n

µ IndHn,1 H (Sµ) 1

as Hn,1-module (permutation module).

r > 1 Mµ := mµ · Hn,r

(mµ ∈ Hn,r) “ permutation module”

Sn,r := EndHn,r ( ⊕

µ Mµ)

; cyclotomic q-Schur alg. Today Introduce an algebra U ass. to Cartan data of glm s.t. U ↠ Sn,r (Drinfeld type presentation). Representation theory of U and Sn,r.

highest weight modules. Harish-Chandra Ind and Res (UL ֒→ UP ֒→ U) evaluation functor Og → UL -mod (Og ⊂ Uq(g) -mod).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 3 / 25

slide-16
SLIDE 16

. . . . . .

Introduction

r = 1 ( Sn,1 EndHn,1(V⊗n) ) V⊗n = ⊕

µ V⊗n µ

: weight space decom. as Uq(glm)-module.

V⊗n

µ IndHn,1 H (Sµ) 1

as Hn,1-module (permutation module).

r > 1 Mµ := mµ · Hn,r

(mµ ∈ Hn,r) “ permutation module”

Sn,r := EndHn,r ( ⊕

µ Mµ)

; cyclotomic q-Schur alg. Today Introduce an algebra U ass. to Cartan data of glm s.t. U ↠ Sn,r (Drinfeld type presentation). Representation theory of U and Sn,r.

highest weight modules. Harish-Chandra Ind and Res (UL ֒→ UP ֒→ U) evaluation functor Og → UL -mod (Og ⊂ Uq(g) -mod).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 3 / 25

slide-17
SLIDE 17

. . . . . .

Cyclotomic q-Schur algebras

Hn,r : Ariki-Koike alg. over Q(q) ass. to Sn ⋉ (Z/rZ)n.

generators: T0, T1, . . . , Tn−1. defining relations:

(T0 − qc1)(T0 − qc2) . . . (T0 − qcr) = 0 (c1, . . . , cr ∈ Z), (Ti − q)(Ti + q−1) = 0 (1 ≤ i ≤ n − 1), + braid relations Li := Ti−1 . . . T1T0T1 . . . Ti−1 (1 ≤ i ≤ n) : Jucys-Murphy elements. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r: Sn,r := EndHn,r ( ⊕

µ∈Λn,r(m)

Mµ) ,

where Mµ = mµ · Hn,r (mµ ∈ Hn,r).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 4 / 25

slide-18
SLIDE 18

. . . . . .

Cyclotomic q-Schur algebras

Hn,r : Ariki-Koike alg. over Q(q) ass. to Sn ⋉ (Z/rZ)n.

generators: T0, T1, . . . , Tn−1. defining relations:

(T0 − qc1)(T0 − qc2) . . . (T0 − qcr) = 0 (c1, . . . , cr ∈ Z), (Ti − q)(Ti + q−1) = 0 (1 ≤ i ≤ n − 1), + braid relations Li := Ti−1 . . . T1T0T1 . . . Ti−1 (1 ≤ i ≤ n) : Jucys-Murphy elements. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r: Sn,r := EndHn,r ( ⊕

µ∈Λn,r(m)

Mµ) ,

where Mµ = mµ · Hn,r (mµ ∈ Hn,r).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 4 / 25

slide-19
SLIDE 19

. . . . . .

Cyclotomic q-Schur algebras

Hn,r : Ariki-Koike alg. over Q(q) ass. to Sn ⋉ (Z/rZ)n.

generators: T0, T1, . . . , Tn−1. defining relations:

(T0 − qc1)(T0 − qc2) . . . (T0 − qcr) = 0 (c1, . . . , cr ∈ Z), (Ti − q)(Ti + q−1) = 0 (1 ≤ i ≤ n − 1), + braid relations Li := Ti−1 . . . T1T0T1 . . . Ti−1 (1 ≤ i ≤ n) : Jucys-Murphy elements. Sn,r : cyclotomic q-Schur algebra ass. to Hn,r: Sn,r := EndHn,r ( ⊕

µ∈Λn,r(m)

Mµ) ,

where Mµ = mµ · Hn,r (mµ ∈ Hn,r).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 4 / 25

slide-20
SLIDE 20

. . . . . .

Borel subalgebras

r = 1 ei, fi (1 ≤ i ≤ m − 1), K±

j ,(1 ≤ j ≤ m) : Chevalley gen. of Uq(glm).

Recall ρ : Uq(glm) ↠ Sn,1.

U≥0

q := ⟨ei, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

U≤0

q := ⟨ fi, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

Sn,1 = ρ(U≤0

q

) · ρ(U≥0

q

)

.

Theorem (Du-Rui)

. .

∃S ≥0 n,r , ∃S ≤0 n,r ⊂alg. Sn,r s.t. Sn,r = S ≤0 n,r · S ≥0 n,r .

Moreover, S ≤0

n,r ρ(U≤0 q

), S ≥0

n,r ρ(U≥0 q

)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 5 / 25

slide-21
SLIDE 21

. . . . . .

Borel subalgebras

r = 1 ei, fi (1 ≤ i ≤ m − 1), K±

j ,(1 ≤ j ≤ m) : Chevalley gen. of Uq(glm).

Recall ρ : Uq(glm) ↠ Sn,1.

U≥0

q := ⟨ei, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

U≤0

q := ⟨ fi, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

Sn,1 = ρ(U≤0

q

) · ρ(U≥0

q

)

.

Theorem (Du-Rui)

. .

∃S ≥0 n,r , ∃S ≤0 n,r ⊂alg. Sn,r s.t. Sn,r = S ≤0 n,r · S ≥0 n,r .

Moreover, S ≤0

n,r ρ(U≤0 q

), S ≥0

n,r ρ(U≥0 q

)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 5 / 25

slide-22
SLIDE 22

. . . . . .

Borel subalgebras

r = 1 ei, fi (1 ≤ i ≤ m − 1), K±

j ,(1 ≤ j ≤ m) : Chevalley gen. of Uq(glm).

Recall ρ : Uq(glm) ↠ Sn,1.

U≥0

q := ⟨ei, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

U≤0

q := ⟨ fi, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

Sn,1 = ρ(U≤0

q

) · ρ(U≥0

q

)

.

Theorem (Du-Rui)

. .

∃S ≥0 n,r , ∃S ≤0 n,r ⊂alg. Sn,r s.t. Sn,r = S ≤0 n,r · S ≥0 n,r .

Moreover, S ≤0

n,r ρ(U≤0 q

), S ≥0

n,r ρ(U≥0 q

)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 5 / 25

slide-23
SLIDE 23

. . . . . .

Borel subalgebras

r = 1 ei, fi (1 ≤ i ≤ m − 1), K±

j ,(1 ≤ j ≤ m) : Chevalley gen. of Uq(glm).

Recall ρ : Uq(glm) ↠ Sn,1.

U≥0

q := ⟨ei, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

U≤0

q := ⟨ fi, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

Sn,1 = ρ(U≤0

q

) · ρ(U≥0

q

)

.

Theorem (Du-Rui)

. .

∃S ≥0 n,r , ∃S ≤0 n,r ⊂alg. Sn,r s.t. Sn,r = S ≤0 n,r · S ≥0 n,r .

Moreover, S ≤0

n,r ρ(U≤0 q

), S ≥0

n,r ρ(U≥0 q

)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 5 / 25

slide-24
SLIDE 24

. . . . . .

Borel subalgebras

r = 1 ei, fi (1 ≤ i ≤ m − 1), K±

j ,(1 ≤ j ≤ m) : Chevalley gen. of Uq(glm).

Recall ρ : Uq(glm) ↠ Sn,1.

U≥0

q := ⟨ei, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

U≤0

q := ⟨ fi, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

Sn,1 = ρ(U≤0

q

) · ρ(U≥0

q

)

.

Theorem (Du-Rui)

. .

∃S ≥0 n,r , ∃S ≤0 n,r ⊂alg. Sn,r s.t. Sn,r = S ≤0 n,r · S ≥0 n,r .

Moreover, S ≤0

n,r ρ(U≤0 q

), S ≥0

n,r ρ(U≥0 q

)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 5 / 25

slide-25
SLIDE 25

. . . . . .

Borel subalgebras

r = 1 ei, fi (1 ≤ i ≤ m − 1), K±

j ,(1 ≤ j ≤ m) : Chevalley gen. of Uq(glm).

Recall ρ : Uq(glm) ↠ Sn,1.

U≥0

q := ⟨ei, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

U≤0

q := ⟨ fi, K± j

  • 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m⟩
  • alg. ⊂ Uq(glm)

Sn,1 = ρ(U≤0

q

) · ρ(U≥0

q

)

.

Theorem (Du-Rui)

. .

∃S ≥0 n,r , ∃S ≤0 n,r ⊂alg. Sn,r s.t. Sn,r = S ≤0 n,r · S ≥0 n,r .

Moreover, S ≤0

n,r ρ(U≤0 q

), S ≥0

n,r ρ(U≥0 q

)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 5 / 25

slide-26
SLIDE 26

. . . . . .

Cartan data

m = (m1, . . . , mr) ∈ Zr s.t. mk ≥ n. Put m = m1 + · · · + mr. {1, 2, . . . , m}

1:1

←→ Γ(m) := {(i, k)

  • 1 ≤ i ≤ mk, 1 ≤ k ≤ r}

∈ ∈ ∑k−1

l=1 ml + i ←→ (i, k)

{1, 2, . . . , m − 1}

1:1

←→ Γ′(m) := Γ(m) \ {(mr, r)} P := ⊕m

i=1 Zεi =

⊕

(i,k)∈Γ(m) Zε(i,k) : weight lattice of glm.

Q := ⊕m−1

i=1 Zαi =

⊕

(i,k)∈Γ′(m) Zα(i,k) : root lattice of glm.

(αi = εi − εi+1 : simple root) For (i, k), (j, l) ∈ Γ(m), put

a(i,k)(j,l) =              1

if (j, l) = (i, k)

−1

if (j, l) = (i + 1, k)

  • therwise

(note (mk + 1, k) = (1, k + 1))

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 6 / 25

slide-27
SLIDE 27

. . . . . .

Cartan data

m = (m1, . . . , mr) ∈ Zr s.t. mk ≥ n. Put m = m1 + · · · + mr. {1, 2, . . . , m}

1:1

←→ Γ(m) := {(i, k)

  • 1 ≤ i ≤ mk, 1 ≤ k ≤ r}

∈ ∈ ∑k−1

l=1 ml + i ←→ (i, k)

{1, 2, . . . , m − 1}

1:1

←→ Γ′(m) := Γ(m) \ {(mr, r)} P := ⊕m

i=1 Zεi =

⊕

(i,k)∈Γ(m) Zε(i,k) : weight lattice of glm.

Q := ⊕m−1

i=1 Zαi =

⊕

(i,k)∈Γ′(m) Zα(i,k) : root lattice of glm.

(αi = εi − εi+1 : simple root) For (i, k), (j, l) ∈ Γ(m), put

a(i,k)(j,l) =              1

if (j, l) = (i, k)

−1

if (j, l) = (i + 1, k)

  • therwise

(note (mk + 1, k) = (1, k + 1))

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 6 / 25

slide-28
SLIDE 28

. . . . . .

Cartan data

m = (m1, . . . , mr) ∈ Zr s.t. mk ≥ n. Put m = m1 + · · · + mr. {1, 2, . . . , m}

1:1

←→ Γ(m) := {(i, k)

  • 1 ≤ i ≤ mk, 1 ≤ k ≤ r}

∈ ∈ ∑k−1

l=1 ml + i ←→ (i, k)

{1, 2, . . . , m − 1}

1:1

←→ Γ′(m) := Γ(m) \ {(mr, r)} P := ⊕m

i=1 Zεi =

⊕

(i,k)∈Γ(m) Zε(i,k) : weight lattice of glm.

Q := ⊕m−1

i=1 Zαi =

⊕

(i,k)∈Γ′(m) Zα(i,k) : root lattice of glm.

(αi = εi − εi+1 : simple root) For (i, k), (j, l) ∈ Γ(m), put

a(i,k)(j,l) =              1

if (j, l) = (i, k)

−1

if (j, l) = (i + 1, k)

  • therwise

(note (mk + 1, k) = (1, k + 1))

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 6 / 25

slide-29
SLIDE 29

. . . . . .

Cartan data

m = (m1, . . . , mr) ∈ Zr s.t. mk ≥ n. Put m = m1 + · · · + mr. {1, 2, . . . , m}

1:1

←→ Γ(m) := {(i, k)

  • 1 ≤ i ≤ mk, 1 ≤ k ≤ r}

∈ ∈ ∑k−1

l=1 ml + i ←→ (i, k)

{1, 2, . . . , m − 1}

1:1

←→ Γ′(m) := Γ(m) \ {(mr, r)} P := ⊕m

i=1 Zεi =

⊕

(i,k)∈Γ(m) Zε(i,k) : weight lattice of glm.

Q := ⊕m−1

i=1 Zαi =

⊕

(i,k)∈Γ′(m) Zα(i,k) : root lattice of glm.

(αi = εi − εi+1 : simple root) For (i, k), (j, l) ∈ Γ(m), put

a(i,k)(j,l) =              1

if (j, l) = (i, k)

−1

if (j, l) = (i + 1, k)

  • therwise

(note (mk + 1, k) = (1, k + 1))

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 6 / 25

slide-30
SLIDE 30

. . . . . .

Cartan data

m = (m1, . . . , mr) ∈ Zr s.t. mk ≥ n. Put m = m1 + · · · + mr. {1, 2, . . . , m}

1:1

←→ Γ(m) := {(i, k)

  • 1 ≤ i ≤ mk, 1 ≤ k ≤ r}

∈ ∈ ∑k−1

l=1 ml + i ←→ (i, k)

{1, 2, . . . , m − 1}

1:1

←→ Γ′(m) := Γ(m) \ {(mr, r)} P := ⊕m

i=1 Zεi =

⊕

(i,k)∈Γ(m) Zε(i,k) : weight lattice of glm.

Q := ⊕m−1

i=1 Zαi =

⊕

(i,k)∈Γ′(m) Zα(i,k) : root lattice of glm.

(αi = εi − εi+1 : simple root) For (i, k), (j, l) ∈ Γ(m), put

a(i,k)(j,l) =              1

if (j, l) = (i, k)

−1

if (j, l) = (i + 1, k)

  • therwise

(note (mk + 1, k) = (1, k + 1))

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 6 / 25

slide-31
SLIDE 31

. . . . . .

Cartan data

m = (m1, . . . , mr) ∈ Zr s.t. mk ≥ n. Put m = m1 + · · · + mr. {1, 2, . . . , m}

1:1

←→ Γ(m) := {(i, k)

  • 1 ≤ i ≤ mk, 1 ≤ k ≤ r}

∈ ∈ ∑k−1

l=1 ml + i ←→ (i, k)

{1, 2, . . . , m − 1}

1:1

←→ Γ′(m) := Γ(m) \ {(mr, r)} P := ⊕m

i=1 Zεi =

⊕

(i,k)∈Γ(m) Zε(i,k) : weight lattice of glm.

Q := ⊕m−1

i=1 Zαi =

⊕

(i,k)∈Γ′(m) Zα(i,k) : root lattice of glm.

(αi = εi − εi+1 : simple root) For (i, k), (j, l) ∈ Γ(m), put

a(i,k)(j,l) =              1

if (j, l) = (i, k)

−1

if (j, l) = (i + 1, k)

  • therwise

(note (mk + 1, k) = (1, k + 1))

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 6 / 25

slide-32
SLIDE 32

. . . . . .

generators of Sn,r

P≥0 := ⊕

(i,k)∈Γ(m)

Z≥0ε(i,k) ⊂ P Λn,r(m) :=       λ = ∑

(i,k)∈Γ(m)

λ(i,k)ε(i,k) ∈ P≥0

  • ∑

(i,k)∈Γ(m)

λ(i,k) = n       

Denote λ ∈ Λn,r(m) by λ = (λ[1], . . . , λ[r]), where λ[k] = (λ(1,k), λ(2,k), . . . , λ(mk,k)). Recall Sn,r := S ≤0

n,r · S ≥0 n,r

and S ≤0

n,r ρ(U≤0 q ),

S ≥0

n,r ρ(U≥0 q )

ρ : Uq(glm) ↠ Sn,1 = ρ(U≤0

q ) · ρ(U≥0 q ).

Define X±

(i,k),0, K± ( j,l) ∈ Sn,r

((i, k) ∈ Γ′(m), ( j, l) ∈ Γ(m)) by X+

(i,k),0 : image of e(i,k) in S ≥0 n,r ρ(U≥0 q ).

X−

(i,k),0 : image of f(i,k) in S ≤0 n,r ρ(U≤0 q ).

K±

( j,l) : image of K± ( j,l) in S ≥0 n,r ρ(U≥0 q ) or S ≤0 n,r ρ(U≤0 q ).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 7 / 25

slide-33
SLIDE 33

. . . . . .

generators of Sn,r

P≥0 := ⊕

(i,k)∈Γ(m)

Z≥0ε(i,k) ⊂ P Λn,r(m) :=       λ = ∑

(i,k)∈Γ(m)

λ(i,k)ε(i,k) ∈ P≥0

  • ∑

(i,k)∈Γ(m)

λ(i,k) = n       

Denote λ ∈ Λn,r(m) by λ = (λ[1], . . . , λ[r]), where λ[k] = (λ(1,k), λ(2,k), . . . , λ(mk,k)). Recall Sn,r := S ≤0

n,r · S ≥0 n,r

and S ≤0

n,r ρ(U≤0 q ),

S ≥0

n,r ρ(U≥0 q )

ρ : Uq(glm) ↠ Sn,1 = ρ(U≤0

q ) · ρ(U≥0 q ).

Define X±

(i,k),0, K± ( j,l) ∈ Sn,r

((i, k) ∈ Γ′(m), ( j, l) ∈ Γ(m)) by X+

(i,k),0 : image of e(i,k) in S ≥0 n,r ρ(U≥0 q ).

X−

(i,k),0 : image of f(i,k) in S ≤0 n,r ρ(U≤0 q ).

K±

( j,l) : image of K± ( j,l) in S ≥0 n,r ρ(U≥0 q ) or S ≤0 n,r ρ(U≤0 q ).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 7 / 25

slide-34
SLIDE 34

. . . . . .

generators of Sn,r

P≥0 := ⊕

(i,k)∈Γ(m)

Z≥0ε(i,k) ⊂ P Λn,r(m) :=       λ = ∑

(i,k)∈Γ(m)

λ(i,k)ε(i,k) ∈ P≥0

  • ∑

(i,k)∈Γ(m)

λ(i,k) = n       

Denote λ ∈ Λn,r(m) by λ = (λ[1], . . . , λ[r]), where λ[k] = (λ(1,k), λ(2,k), . . . , λ(mk,k)). Recall Sn,r := S ≤0

n,r · S ≥0 n,r

and S ≤0

n,r ρ(U≤0 q ),

S ≥0

n,r ρ(U≥0 q )

ρ : Uq(glm) ↠ Sn,1 = ρ(U≤0

q ) · ρ(U≥0 q ).

Define X±

(i,k),0, K± ( j,l) ∈ Sn,r

((i, k) ∈ Γ′(m), ( j, l) ∈ Γ(m)) by X+

(i,k),0 : image of e(i,k) in S ≥0 n,r ρ(U≥0 q ).

X−

(i,k),0 : image of f(i,k) in S ≤0 n,r ρ(U≤0 q ).

K±

( j,l) : image of K± ( j,l) in S ≥0 n,r ρ(U≥0 q ) or S ≤0 n,r ρ(U≤0 q ).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 7 / 25

slide-35
SLIDE 35

. . . . . .

generators of Sn,r

P≥0 := ⊕

(i,k)∈Γ(m)

Z≥0ε(i,k) ⊂ P Λn,r(m) :=       λ = ∑

(i,k)∈Γ(m)

λ(i,k)ε(i,k) ∈ P≥0

  • ∑

(i,k)∈Γ(m)

λ(i,k) = n       

Denote λ ∈ Λn,r(m) by λ = (λ[1], . . . , λ[r]), where λ[k] = (λ(1,k), λ(2,k), . . . , λ(mk,k)). Recall Sn,r := S ≤0

n,r · S ≥0 n,r

and S ≤0

n,r ρ(U≤0 q ),

S ≥0

n,r ρ(U≥0 q )

ρ : Uq(glm) ↠ Sn,1 = ρ(U≤0

q ) · ρ(U≥0 q ).

Define X±

(i,k),0, K± ( j,l) ∈ Sn,r

((i, k) ∈ Γ′(m), ( j, l) ∈ Γ(m)) by X+

(i,k),0 : image of e(i,k) in S ≥0 n,r ρ(U≥0 q ).

X−

(i,k),0 : image of f(i,k) in S ≤0 n,r ρ(U≤0 q ).

K±

( j,l) : image of K± ( j,l) in S ≥0 n,r ρ(U≥0 q ) or S ≤0 n,r ρ(U≤0 q ).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 7 / 25

slide-36
SLIDE 36

. . . . . .

generators of Sn,r

P≥0 := ⊕

(i,k)∈Γ(m)

Z≥0ε(i,k) ⊂ P Λn,r(m) :=       λ = ∑

(i,k)∈Γ(m)

λ(i,k)ε(i,k) ∈ P≥0

  • ∑

(i,k)∈Γ(m)

λ(i,k) = n       

Denote λ ∈ Λn,r(m) by λ = (λ[1], . . . , λ[r]), where λ[k] = (λ(1,k), λ(2,k), . . . , λ(mk,k)). Recall Sn,r := S ≤0

n,r · S ≥0 n,r

and S ≤0

n,r ρ(U≤0 q ),

S ≥0

n,r ρ(U≥0 q )

ρ : Uq(glm) ↠ Sn,1 = ρ(U≤0

q ) · ρ(U≥0 q ).

Define X±

(i,k),0, K± ( j,l) ∈ Sn,r

((i, k) ∈ Γ′(m), ( j, l) ∈ Γ(m)) by X+

(i,k),0 : image of e(i,k) in S ≥0 n,r ρ(U≥0 q ).

X−

(i,k),0 : image of f(i,k) in S ≤0 n,r ρ(U≤0 q ).

K±

( j,l) : image of K± ( j,l) in S ≥0 n,r ρ(U≥0 q ) or S ≤0 n,r ρ(U≤0 q ).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 7 / 25

slide-37
SLIDE 37

. . . . . .

generators of Sn,r

P≥0 := ⊕

(i,k)∈Γ(m)

Z≥0ε(i,k) ⊂ P Λn,r(m) :=       λ = ∑

(i,k)∈Γ(m)

λ(i,k)ε(i,k) ∈ P≥0

  • ∑

(i,k)∈Γ(m)

λ(i,k) = n       

Denote λ ∈ Λn,r(m) by λ = (λ[1], . . . , λ[r]), where λ[k] = (λ(1,k), λ(2,k), . . . , λ(mk,k)). Recall Sn,r := S ≤0

n,r · S ≥0 n,r

and S ≤0

n,r ρ(U≤0 q ),

S ≥0

n,r ρ(U≥0 q )

ρ : Uq(glm) ↠ Sn,1 = ρ(U≤0

q ) · ρ(U≥0 q ).

Define X±

(i,k),0, K± ( j,l) ∈ Sn,r

((i, k) ∈ Γ′(m), ( j, l) ∈ Γ(m)) by X+

(i,k),0 : image of e(i,k) in S ≥0 n,r ρ(U≥0 q ).

X−

(i,k),0 : image of f(i,k) in S ≤0 n,r ρ(U≤0 q ).

K±

( j,l) : image of K± ( j,l) in S ≥0 n,r ρ(U≥0 q ) or S ≤0 n,r ρ(U≤0 q ).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 7 / 25

slide-38
SLIDE 38

. . . . . .

generators of Sn,r

P≥0 := ⊕

(i,k)∈Γ(m)

Z≥0ε(i,k) ⊂ P Λn,r(m) :=       λ = ∑

(i,k)∈Γ(m)

λ(i,k)ε(i,k) ∈ P≥0

  • ∑

(i,k)∈Γ(m)

λ(i,k) = n       

Denote λ ∈ Λn,r(m) by λ = (λ[1], . . . , λ[r]), where λ[k] = (λ(1,k), λ(2,k), . . . , λ(mk,k)). Recall Sn,r := S ≤0

n,r · S ≥0 n,r

and S ≤0

n,r ρ(U≤0 q ),

S ≥0

n,r ρ(U≥0 q )

ρ : Uq(glm) ↠ Sn,1 = ρ(U≤0

q ) · ρ(U≥0 q ).

Define X±

(i,k),0, K± ( j,l) ∈ Sn,r

((i, k) ∈ Γ′(m), ( j, l) ∈ Γ(m)) by X+

(i,k),0 : image of e(i,k) in S ≥0 n,r ρ(U≥0 q ).

X−

(i,k),0 : image of f(i,k) in S ≤0 n,r ρ(U≤0 q ).

K±

( j,l) : image of K± ( j,l) in S ≥0 n,r ρ(U≥0 q ) or S ≤0 n,r ρ(U≤0 q ).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 7 / 25

slide-39
SLIDE 39

. . . . . .

generators of Sn,r

P≥0 := ⊕

(i,k)∈Γ(m)

Z≥0ε(i,k) ⊂ P Λn,r(m) :=       λ = ∑

(i,k)∈Γ(m)

λ(i,k)ε(i,k) ∈ P≥0

  • ∑

(i,k)∈Γ(m)

λ(i,k) = n       

Denote λ ∈ Λn,r(m) by λ = (λ[1], . . . , λ[r]), where λ[k] = (λ(1,k), λ(2,k), . . . , λ(mk,k)). Recall Sn,r := S ≤0

n,r · S ≥0 n,r

and S ≤0

n,r ρ(U≤0 q ),

S ≥0

n,r ρ(U≥0 q )

ρ : Uq(glm) ↠ Sn,1 = ρ(U≤0

q ) · ρ(U≥0 q ).

Define X±

(i,k),0, K± ( j,l) ∈ Sn,r

((i, k) ∈ Γ′(m), ( j, l) ∈ Γ(m)) by X+

(i,k),0 : image of e(i,k) in S ≥0 n,r ρ(U≥0 q ).

X−

(i,k),0 : image of f(i,k) in S ≤0 n,r ρ(U≤0 q ).

K±

( j,l) : image of K± ( j,l) in S ≥0 n,r ρ(U≥0 q ) or S ≤0 n,r ρ(U≤0 q ).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 7 / 25

slide-40
SLIDE 40

. . . . . .

Symmetric polynomials Φ±

t

Define Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk] (t ≥ 1) by

Φ±

1(X1, . . . , Xk) := X1 + X2 + · · · + Xk.

Φ±

t+1(X1, . . . , Xk)

:= Xt+1

1

+

k

∑

s=2

( Φ±

t (X1, . . . , Xs)Xs − q∓2Φ± t (X1, . . . , Xs−1)Xs

) ( Z[q, q−1][X1, . . . , Xs] ֒→ Z[q, q−1][X1, . . . , Xk], Xi → Xi )

.

Lemma

. .

Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk]Sk.

.

Remark

. .

Φ±

t (X1, . . . , Xk−1, 0) = Φ± t (X1, . . . , Xk−1)

∃Φ±

t (X1, X2, . . . ) : symmetric function

s.t. Φ±

t (X1, . . . , Xk, 0, . . . ) = Φ± t (X1, . . . , Xk)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 8 / 25

slide-41
SLIDE 41

. . . . . .

Symmetric polynomials Φ±

t

Define Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk] (t ≥ 1) by

Φ±

1(X1, . . . , Xk) := X1 + X2 + · · · + Xk.

Φ±

t+1(X1, . . . , Xk)

:= Xt+1

1

+

k

∑

s=2

( Φ±

t (X1, . . . , Xs)Xs − q∓2Φ± t (X1, . . . , Xs−1)Xs

) ( Z[q, q−1][X1, . . . , Xs] ֒→ Z[q, q−1][X1, . . . , Xk], Xi → Xi )

.

Lemma

. .

Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk]Sk.

.

Remark

. .

Φ±

t (X1, . . . , Xk−1, 0) = Φ± t (X1, . . . , Xk−1)

∃Φ±

t (X1, X2, . . . ) : symmetric function

s.t. Φ±

t (X1, . . . , Xk, 0, . . . ) = Φ± t (X1, . . . , Xk)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 8 / 25

slide-42
SLIDE 42

. . . . . .

Symmetric polynomials Φ±

t

Define Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk] (t ≥ 1) by

Φ±

1(X1, . . . , Xk) := X1 + X2 + · · · + Xk.

Φ±

t+1(X1, . . . , Xk)

:= Xt+1

1

+

k

∑

s=2

( Φ±

t (X1, . . . , Xs)Xs − q∓2Φ± t (X1, . . . , Xs−1)Xs

) ( Z[q, q−1][X1, . . . , Xs] ֒→ Z[q, q−1][X1, . . . , Xk], Xi → Xi )

.

Lemma

. .

Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk]Sk.

.

Remark

. .

Φ±

t (X1, . . . , Xk−1, 0) = Φ± t (X1, . . . , Xk−1)

∃Φ±

t (X1, X2, . . . ) : symmetric function

s.t. Φ±

t (X1, . . . , Xk, 0, . . . ) = Φ± t (X1, . . . , Xk)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 8 / 25

slide-43
SLIDE 43

. . . . . .

Symmetric polynomials Φ±

t

Define Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk] (t ≥ 1) by

Φ±

1(X1, . . . , Xk) := X1 + X2 + · · · + Xk.

Φ±

t+1(X1, . . . , Xk)

:= Xt+1

1

+

k

∑

s=2

( Φ±

t (X1, . . . , Xs)Xs − q∓2Φ± t (X1, . . . , Xs−1)Xs

) ( Z[q, q−1][X1, . . . , Xs] ֒→ Z[q, q−1][X1, . . . , Xk], Xi → Xi )

.

Lemma

. .

Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk]Sk.

.

Remark

. .

Φ±

t (X1, . . . , Xk−1, 0) = Φ± t (X1, . . . , Xk−1)

∃Φ±

t (X1, X2, . . . ) : symmetric function

s.t. Φ±

t (X1, . . . , Xk, 0, . . . ) = Φ± t (X1, . . . , Xk)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 8 / 25

slide-44
SLIDE 44

. . . . . .

Symmetric polynomials Φ±

t

Define Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk] (t ≥ 1) by

Φ±

1(X1, . . . , Xk) := X1 + X2 + · · · + Xk.

Φ±

t+1(X1, . . . , Xk)

:= Xt+1

1

+

k

∑

s=2

( Φ±

t (X1, . . . , Xs)Xs − q∓2Φ± t (X1, . . . , Xs−1)Xs

) ( Z[q, q−1][X1, . . . , Xs] ֒→ Z[q, q−1][X1, . . . , Xk], Xi → Xi )

.

Lemma

. .

Φ±

t (X1, . . . , Xk) ∈ Z[q, q−1][X1, . . . , Xk]Sk.

.

Remark

. .

Φ±

t (X1, . . . , Xk−1, 0) = Φ± t (X1, . . . , Xk−1)

∃Φ±

t (X1, X2, . . . ) : symmetric function

s.t. Φ±

t (X1, . . . , Xk, 0, . . . ) = Φ± t (X1, . . . , Xk)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 8 / 25

slide-45
SLIDE 45

. . . . . .

generators of Sn,r We already defined X±

(i,k),0, K± ( j,l) ∈ Sn,r

( (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m) ).

Recall Sn,r = EndHn,r

( ⊕

µ∈Λn,r(m) mµ · Hn,r

)

. Define H±

( j,l),t ∈ Sn,r

((j, l) ∈ Γ(m), t ≥ 1) by H+

( j,l),t(mµ) :=

1 q−t+1(q − q−1)t−1mµΦ+

t (LN, LN−1, . . . , LN−µ(l)

j +1),

H−

( j,l),t(mµ) :=

−1 qt−1(q − q−1)t−1mµΦ−

t (LN, LN−1, . . . , LN−µ(l)

j +1),

where N = ∑l−1

s=1 |µ[s]| + ∑j i=1 µ(i,l).

Define X±

(i,k),t ∈ Sn,r

((i, k) ∈ Γ′(m), t ≥ 1) by X±

(i,k),t+1 :=

1 q − q−1 ( H+

(i,k),1X± (i,k),t − X± (i,k),tH+ (i,k),1

)

.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 9 / 25

slide-46
SLIDE 46

. . . . . .

generators of Sn,r We already defined X±

(i,k),0, K± ( j,l) ∈ Sn,r

( (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m) ).

Recall Sn,r = EndHn,r

( ⊕

µ∈Λn,r(m) mµ · Hn,r

)

. Define H±

( j,l),t ∈ Sn,r

((j, l) ∈ Γ(m), t ≥ 1) by H+

( j,l),t(mµ) :=

1 q−t+1(q − q−1)t−1mµΦ+

t (LN, LN−1, . . . , LN−µ(l)

j +1),

H−

( j,l),t(mµ) :=

−1 qt−1(q − q−1)t−1mµΦ−

t (LN, LN−1, . . . , LN−µ(l)

j +1),

where N = ∑l−1

s=1 |µ[s]| + ∑j i=1 µ(i,l).

Define X±

(i,k),t ∈ Sn,r

((i, k) ∈ Γ′(m), t ≥ 1) by X±

(i,k),t+1 :=

1 q − q−1 ( H+

(i,k),1X± (i,k),t − X± (i,k),tH+ (i,k),1

)

.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 9 / 25

slide-47
SLIDE 47

. . . . . .

generators of Sn,r We already defined X±

(i,k),0, K± ( j,l) ∈ Sn,r

( (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m) ).

Recall Sn,r = EndHn,r

( ⊕

µ∈Λn,r(m) mµ · Hn,r

)

. Define H±

( j,l),t ∈ Sn,r

((j, l) ∈ Γ(m), t ≥ 1) by H+

( j,l),t(mµ) :=

1 q−t+1(q − q−1)t−1mµΦ+

t (LN, LN−1, . . . , LN−µ(l)

j +1),

H−

( j,l),t(mµ) :=

−1 qt−1(q − q−1)t−1mµΦ−

t (LN, LN−1, . . . , LN−µ(l)

j +1),

where N = ∑l−1

s=1 |µ[s]| + ∑j i=1 µ(i,l).

Define X±

(i,k),t ∈ Sn,r

((i, k) ∈ Γ′(m), t ≥ 1) by X±

(i,k),t+1 :=

1 q − q−1 ( H+

(i,k),1X± (i,k),t − X± (i,k),tH+ (i,k),1

)

.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 9 / 25

slide-48
SLIDE 48

. . . . . .

generators of Sn,r We already defined X±

(i,k),0, K± ( j,l) ∈ Sn,r

( (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m) ).

Recall Sn,r = EndHn,r

( ⊕

µ∈Λn,r(m) mµ · Hn,r

)

. Define H±

( j,l),t ∈ Sn,r

((j, l) ∈ Γ(m), t ≥ 1) by H+

( j,l),t(mµ) :=

1 q−t+1(q − q−1)t−1mµΦ+

t (LN, LN−1, . . . , LN−µ(l)

j +1),

H−

( j,l),t(mµ) :=

−1 qt−1(q − q−1)t−1mµΦ−

t (LN, LN−1, . . . , LN−µ(l)

j +1),

where N = ∑l−1

s=1 |µ[s]| + ∑j i=1 µ(i,l).

Define X±

(i,k),t ∈ Sn,r

((i, k) ∈ Γ′(m), t ≥ 1) by X±

(i,k),t+1 :=

1 q − q−1 ( H+

(i,k),1X± (i,k),t − X± (i,k),tH+ (i,k),1

)

.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 9 / 25

slide-49
SLIDE 49

. . . . . .

generators of Sn,r We already defined X±

(i,k),0, K± ( j,l) ∈ Sn,r

( (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m) ).

Recall Sn,r = EndHn,r

( ⊕

µ∈Λn,r(m) mµ · Hn,r

)

. Define H±

( j,l),t ∈ Sn,r

((j, l) ∈ Γ(m), t ≥ 1) by H+

( j,l),t(mµ) :=

1 q−t+1(q − q−1)t−1mµΦ+

t (LN, LN−1, . . . , LN−µ(l)

j +1),

H−

( j,l),t(mµ) :=

−1 qt−1(q − q−1)t−1mµΦ−

t (LN, LN−1, . . . , LN−µ(l)

j +1),

where N = ∑l−1

s=1 |µ[s]| + ∑j i=1 µ(i,l).

Define X±

(i,k),t ∈ Sn,r

((i, k) ∈ Γ′(m), t ≥ 1) by X±

(i,k),t+1 :=

1 q − q−1 ( H+

(i,k),1X± (i,k),t − X± (i,k),tH+ (i,k),1

)

.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 9 / 25

slide-50
SLIDE 50

. . . . . .

Algebra U .

Definition

. .

m = (m1, . . . , mr) ∈ Zr

>0.

U : associative algebra over Q(q) defined by

generators: X±

(i,k),t, K± (j,l), H± ( j,l),t, Ck

((i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r)

defining relations:

Ck : central elements, K+

( j,l)K− ( j,l) = K− ( j,l)K+ ( j,l) = 1,

H±

(i,k),0 = 1,

[Kε

(i,k), Kε′ ( j,l)] = [Kε (i,k), Hε′ ( j,l),s] = [Hε (i,k),t, Hε′ ( j,l),s] = 0

(ε, ε′ ∈ {+, −}) K( j,l)X±

(i,k),tK− ( j,l) = q±a(i,k)( j,l)X± (i,k),t,

[H+

( j,l),s+1, X± (i,k),t] = q±a(i,k)( j,l)H+ ( j,l),sX± (i,k),t+1 − q∓a(i,k)( j,l)X± (i,k),t+1H+ ( j,l),s

[H−

( j,l),s+1, X± (i,k),t] = q∓a(i,k)( j,l)H− ( j,l),sX± (i,k),t+1 − q±a(i,k)( j,l)X± (i,k),t+1H− ( j,l),s

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 10 / 25

slide-51
SLIDE 51

. . . . . .

Algebra U .

Definition

. .

m = (m1, . . . , mr) ∈ Zr

>0.

U : associative algebra over Q(q) defined by

generators: X±

(i,k),t, K± (j,l), H± ( j,l),t, Ck

((i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r)

defining relations:

Ck : central elements, K+

( j,l)K− ( j,l) = K− ( j,l)K+ ( j,l) = 1,

H±

(i,k),0 = 1,

[Kε

(i,k), Kε′ ( j,l)] = [Kε (i,k), Hε′ ( j,l),s] = [Hε (i,k),t, Hε′ ( j,l),s] = 0

(ε, ε′ ∈ {+, −}) K( j,l)X±

(i,k),tK− ( j,l) = q±a(i,k)( j,l)X± (i,k),t,

[H+

( j,l),s+1, X± (i,k),t] = q±a(i,k)( j,l)H+ ( j,l),sX± (i,k),t+1 − q∓a(i,k)( j,l)X± (i,k),t+1H+ ( j,l),s

[H−

( j,l),s+1, X± (i,k),t] = q∓a(i,k)( j,l)H− ( j,l),sX± (i,k),t+1 − q±a(i,k)( j,l)X± (i,k),t+1H− ( j,l),s

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 10 / 25

slide-52
SLIDE 52

. . . . . .

Algebra U .

Definition

. .

m = (m1, . . . , mr) ∈ Zr

>0.

U : associative algebra over Q(q) defined by

generators: X±

(i,k),t, K± (j,l), H± ( j,l),t, Ck

((i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r)

defining relations:

Ck : central elements, K+

( j,l)K− ( j,l) = K− ( j,l)K+ ( j,l) = 1,

H±

(i,k),0 = 1,

[Kε

(i,k), Kε′ ( j,l)] = [Kε (i,k), Hε′ ( j,l),s] = [Hε (i,k),t, Hε′ ( j,l),s] = 0

(ε, ε′ ∈ {+, −}) K( j,l)X±

(i,k),tK− ( j,l) = q±a(i,k)( j,l)X± (i,k),t,

[H+

( j,l),s+1, X± (i,k),t] = q±a(i,k)( j,l)H+ ( j,l),sX± (i,k),t+1 − q∓a(i,k)( j,l)X± (i,k),t+1H+ ( j,l),s

[H−

( j,l),s+1, X± (i,k),t] = q∓a(i,k)( j,l)H− ( j,l),sX± (i,k),t+1 − q±a(i,k)( j,l)X± (i,k),t+1H− ( j,l),s

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 10 / 25

slide-53
SLIDE 53

. . . . . .

Algebra U .

Definition

. .

m = (m1, . . . , mr) ∈ Zr

>0.

U : associative algebra over Q(q) defined by

generators: X±

(i,k),t, K± (j,l), H± ( j,l),t, Ck

((i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r)

defining relations:

Ck : central elements, K+

( j,l)K− ( j,l) = K− ( j,l)K+ ( j,l) = 1,

H±

(i,k),0 = 1,

[Kε

(i,k), Kε′ ( j,l)] = [Kε (i,k), Hε′ ( j,l),s] = [Hε (i,k),t, Hε′ ( j,l),s] = 0

(ε, ε′ ∈ {+, −}) K( j,l)X±

(i,k),tK− ( j,l) = q±a(i,k)( j,l)X± (i,k),t,

[H+

( j,l),s+1, X± (i,k),t] = q±a(i,k)( j,l)H+ ( j,l),sX± (i,k),t+1 − q∓a(i,k)( j,l)X± (i,k),t+1H+ ( j,l),s

[H−

( j,l),s+1, X± (i,k),t] = q∓a(i,k)( j,l)H− ( j,l),sX± (i,k),t+1 − q±a(i,k)( j,l)X± (i,k),t+1H− ( j,l),s

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 10 / 25

slide-54
SLIDE 54

. . . . . .

Algebra U .

Definition

. .

m = (m1, . . . , mr) ∈ Zr

>0.

U : associative algebra over Q(q) defined by

generators: X±

(i,k),t, K± (j,l), H± ( j,l),t, Ck

((i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r)

defining relations:

Ck : central elements, K+

( j,l)K− ( j,l) = K− ( j,l)K+ ( j,l) = 1,

H±

(i,k),0 = 1,

[Kε

(i,k), Kε′ ( j,l)] = [Kε (i,k), Hε′ ( j,l),s] = [Hε (i,k),t, Hε′ ( j,l),s] = 0

(ε, ε′ ∈ {+, −}) K( j,l)X±

(i,k),tK− ( j,l) = q±a(i,k)( j,l)X± (i,k),t,

[H+

( j,l),s+1, X± (i,k),t] = q±a(i,k)( j,l)H+ ( j,l),sX± (i,k),t+1 − q∓a(i,k)( j,l)X± (i,k),t+1H+ ( j,l),s

[H−

( j,l),s+1, X± (i,k),t] = q∓a(i,k)( j,l)H− ( j,l),sX± (i,k),t+1 − q±a(i,k)( j,l)X± (i,k),t+1H− ( j,l),s

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 10 / 25

slide-55
SLIDE 55

. . . . . .

.

defining relations (cont.):

. .

Xε

(i,k),tXε ( j,l),s − Xε ( j,l),sXε (i,k),t = 0

if (j, l) (i, k), (i ± 1, k),

Xε

(i±1,k),0(Xε (i,k),0)2 − (q + q−1)Xε (i,k),0Xε (i±1,0),0Xε (i,k),0 + (Xε (i,k),0)2Xε (i±1,k),0 = 0

[X+

(i,k),t, X− ( j,l),s]

= δ(i,k)(j,l)

                         J+

(i,k),s+t − J− (i,k),s+t

q − q−1

if i mk,

−Ck+1 J+

(mk,k),s+t − J− (mk,k),s+t

q − q−1 + (J+

(mk,k),s+t+1−

J−

(mk,k),s+t+1

)

if i = mk,

where J+

(i,k),0 := K+ (i,k)K− (i+1,k), J− (i,k),0 := K− (i,k)K+ (i+1,k).

J+

(i,k),t := K+ (i,k)K− (i+1,k)

( q−tH+

(i,k),t −

q−1 q − q−1

t−1

∑

h=1

qt−2hH+

(i,k),hH− (i+1,k),t−h

) J−

(i,k),t := K+ (i,k)K− (i+1,k)

( − qtH−

(i+1,k),t −

q q − q−1

t−1

∑

h=1

qt−2hH+

(i,k),hH− (i+1,k),t−h

)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 11 / 25

slide-56
SLIDE 56

. . . . . .

.

defining relations (cont.):

. .

Xε

(i,k),tXε ( j,l),s − Xε ( j,l),sXε (i,k),t = 0

if (j, l) (i, k), (i ± 1, k),

Xε

(i±1,k),0(Xε (i,k),0)2 − (q + q−1)Xε (i,k),0Xε (i±1,0),0Xε (i,k),0 + (Xε (i,k),0)2Xε (i±1,k),0 = 0

[X+

(i,k),t, X− ( j,l),s]

= δ(i,k)(j,l)

                         J+

(i,k),s+t − J− (i,k),s+t

q − q−1

if i mk,

−Ck+1 J+

(mk,k),s+t − J− (mk,k),s+t

q − q−1 + (J+

(mk,k),s+t+1−

J−

(mk,k),s+t+1

)

if i = mk,

where J+

(i,k),0 := K+ (i,k)K− (i+1,k), J− (i,k),0 := K− (i,k)K+ (i+1,k).

J+

(i,k),t := K+ (i,k)K− (i+1,k)

( q−tH+

(i,k),t −

q−1 q − q−1

t−1

∑

h=1

qt−2hH+

(i,k),hH− (i+1,k),t−h

) J−

(i,k),t := K+ (i,k)K− (i+1,k)

( − qtH−

(i+1,k),t −

q q − q−1

t−1

∑

h=1

qt−2hH+

(i,k),hH− (i+1,k),t−h

)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 11 / 25

slide-57
SLIDE 57

. . . . . .

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. There exists a surjective homomorphism

U ↠ Sn,r

s.t. Ck → qck, X±

(i,k),t → X± (i,k),t, K± ( j,l) → K± ( j,l), H± ( j,l),t → H± ( j,l),t.

.

Proposition

. . .

1

There exists a surjective homomorphism

U ↠ Uq(glm)

s.t. Ck → −1, X+

(i,k),0 → e(i,k), X− (i,k),0 → f(i,k), K± ( j,l) → K± ( j,l),

X±

(i,k),t, H± ( j,l),t → 0 (t ≥ 1).

. .

2

There exists a injective homomorphism

Uq(g) ֒→ U

s.t. e(i,k) → X+

(i,k),0, f(i,k) → X− (i,k),0, K± ( j,l) → K± ( j,l).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 12 / 25

slide-58
SLIDE 58

. . . . . .

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. There exists a surjective homomorphism

U ↠ Sn,r

s.t. Ck → qck, X±

(i,k),t → X± (i,k),t, K± ( j,l) → K± ( j,l), H± ( j,l),t → H± ( j,l),t.

.

Proposition

. . .

1

There exists a surjective homomorphism

U ↠ Uq(glm)

s.t. Ck → −1, X+

(i,k),0 → e(i,k), X− (i,k),0 → f(i,k), K± ( j,l) → K± ( j,l),

X±

(i,k),t, H± ( j,l),t → 0 (t ≥ 1).

. .

2

There exists a injective homomorphism

Uq(g) ֒→ U

s.t. e(i,k) → X+

(i,k),0, f(i,k) → X− (i,k),0, K± ( j,l) → K± ( j,l).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 12 / 25

slide-59
SLIDE 59

. . . . . .

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. There exists a surjective homomorphism

U ↠ Sn,r

s.t. Ck → qck, X±

(i,k),t → X± (i,k),t, K± ( j,l) → K± ( j,l), H± ( j,l),t → H± ( j,l),t.

.

Proposition

. . .

1

There exists a surjective homomorphism

U ↠ Uq(glm)

s.t. Ck → −1, X+

(i,k),0 → e(i,k), X− (i,k),0 → f(i,k), K± ( j,l) → K± ( j,l),

X±

(i,k),t, H± ( j,l),t → 0 (t ≥ 1).

. .

2

There exists a injective homomorphism

Uq(g) ֒→ U

s.t. e(i,k) → X+

(i,k),0, f(i,k) → X− (i,k),0, K± ( j,l) → K± ( j,l).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 12 / 25

slide-60
SLIDE 60

. . . . . .

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. There exists a surjective homomorphism

U ↠ Sn,r

s.t. Ck → qck, X±

(i,k),t → X± (i,k),t, K± ( j,l) → K± ( j,l), H± ( j,l),t → H± ( j,l),t.

.

Proposition

. . .

1

There exists a surjective homomorphism

U ↠ Uq(glm)

s.t. Ck → −1, X+

(i,k),0 → e(i,k), X− (i,k),0 → f(i,k), K± ( j,l) → K± ( j,l),

X±

(i,k),t, H± ( j,l),t → 0 (t ≥ 1).

. .

2

There exists a injective homomorphism

Uq(g) ֒→ U

s.t. e(i,k) → X+

(i,k),0, f(i,k) → X− (i,k),0, K± ( j,l) → K± ( j,l).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 12 / 25

slide-61
SLIDE 61

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-62
SLIDE 62

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-63
SLIDE 63

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-64
SLIDE 64

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-65
SLIDE 65

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-66
SLIDE 66

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-67
SLIDE 67

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-68
SLIDE 68

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-69
SLIDE 69

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-70
SLIDE 70

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-71
SLIDE 71

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-72
SLIDE 72

. . . . . .

Highest weight modules of U Define subalgebras U0 and U± of U by

U0 := ⟨K±

( j,l), H± ( j,l),t, Ck

  • (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r⟩

alg..

U+ := ⟨X+

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

U− := ⟨X−

(i,k),t

  • (i, k) ∈ Γ′(m), t ≥ 0⟩

alg..

.

Lemma

. .

U = U−U0U+ (U U− ⊗ U0 ⊗ U+ ???)

For λ ∈ P≥0, φ = (φ±

(i,k),t)(i,k)∈Γ(m),t≥1 (φ± (i,k),t ∈ Q(q)) and

c = (c1, . . . , cr) ∈ Zr, U-module V is a highest weight module (of type 1) with h.w. (λ, φ, c)

if ∃v0 ∈ V s.t.

V = U · v0. X+

(i,k),t · vo = 0 for all (i, k) ∈ Γ′(m), t ≥ 0.

K+

(i,k) · v0 = qλ(i,k)v0, H± (i,k),t · v0 = φ± (i,k),tv0, Ck · v0 = qckv0.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 13 / 25

slide-73
SLIDE 73

. . . . . .

Highest weight modules of U For (λ, φ, c), define the Verma module V(λ, φ, c) in usual way. h.w. module with h.w. (λ, φ, c) is a quotient of V(λ, φ, c).

V(λ, φ, c) has the unique simple top L(λ, φ, c).

.

Problem

. . When is L(λ, φ, c) finite dimensional?. .

Lemma

. .

L(λ, φ, c) : finite dim. λ : r-partition. i.e. λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) ≥ 0 for k = 1, . . . , r. ∵) Recall Uq(g) ֒→ U. v0 : h.w. vector of L(λ, φ, c) v0 : h.w. vector with weight λ as Uq(g)-module

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 14 / 25

slide-74
SLIDE 74

. . . . . .

Highest weight modules of U For (λ, φ, c), define the Verma module V(λ, φ, c) in usual way. h.w. module with h.w. (λ, φ, c) is a quotient of V(λ, φ, c).

V(λ, φ, c) has the unique simple top L(λ, φ, c).

.

Problem

. . When is L(λ, φ, c) finite dimensional?. .

Lemma

. .

L(λ, φ, c) : finite dim. λ : r-partition. i.e. λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) ≥ 0 for k = 1, . . . , r. ∵) Recall Uq(g) ֒→ U. v0 : h.w. vector of L(λ, φ, c) v0 : h.w. vector with weight λ as Uq(g)-module

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 14 / 25

slide-75
SLIDE 75

. . . . . .

Highest weight modules of U For (λ, φ, c), define the Verma module V(λ, φ, c) in usual way. h.w. module with h.w. (λ, φ, c) is a quotient of V(λ, φ, c).

V(λ, φ, c) has the unique simple top L(λ, φ, c).

.

Problem

. . When is L(λ, φ, c) finite dimensional?. .

Lemma

. .

L(λ, φ, c) : finite dim. λ : r-partition. i.e. λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) ≥ 0 for k = 1, . . . , r. ∵) Recall Uq(g) ֒→ U. v0 : h.w. vector of L(λ, φ, c) v0 : h.w. vector with weight λ as Uq(g)-module

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 14 / 25

slide-76
SLIDE 76

. . . . . .

Highest weight modules of U For (λ, φ, c), define the Verma module V(λ, φ, c) in usual way. h.w. module with h.w. (λ, φ, c) is a quotient of V(λ, φ, c).

V(λ, φ, c) has the unique simple top L(λ, φ, c).

.

Problem

. . When is L(λ, φ, c) finite dimensional?. .

Lemma

. .

L(λ, φ, c) : finite dim. λ : r-partition. i.e. λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) ≥ 0 for k = 1, . . . , r. ∵) Recall Uq(g) ֒→ U. v0 : h.w. vector of L(λ, φ, c) v0 : h.w. vector with weight λ as Uq(g)-module

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 14 / 25

slide-77
SLIDE 77

. . . . . .

Highest weight modules of U For (λ, φ, c), define the Verma module V(λ, φ, c) in usual way. h.w. module with h.w. (λ, φ, c) is a quotient of V(λ, φ, c).

V(λ, φ, c) has the unique simple top L(λ, φ, c).

.

Problem

. . When is L(λ, φ, c) finite dimensional?. .

Lemma

. .

L(λ, φ, c) : finite dim. λ : r-partition. i.e. λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) ≥ 0 for k = 1, . . . , r. ∵) Recall Uq(g) ֒→ U. v0 : h.w. vector of L(λ, φ, c) v0 : h.w. vector with weight λ as Uq(g)-module

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 14 / 25

slide-78
SLIDE 78

. . . . . .

Highest weight modules of U For (λ, φ, c), define the Verma module V(λ, φ, c) in usual way. h.w. module with h.w. (λ, φ, c) is a quotient of V(λ, φ, c).

V(λ, φ, c) has the unique simple top L(λ, φ, c).

.

Problem

. . When is L(λ, φ, c) finite dimensional?. .

Lemma

. .

L(λ, φ, c) : finite dim. λ : r-partition. i.e. λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) ≥ 0 for k = 1, . . . , r. ∵) Recall Uq(g) ֒→ U. v0 : h.w. vector of L(λ, φ, c) v0 : h.w. vector with weight λ as Uq(g)-module

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 14 / 25

slide-79
SLIDE 79

. . . . . .

Highest weight modules of U For (λ, φ, c), define the Verma module V(λ, φ, c) in usual way. h.w. module with h.w. (λ, φ, c) is a quotient of V(λ, φ, c).

V(λ, φ, c) has the unique simple top L(λ, φ, c).

.

Problem

. . When is L(λ, φ, c) finite dimensional?. .

Lemma

. .

L(λ, φ, c) : finite dim. λ : r-partition. i.e. λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) ≥ 0 for k = 1, . . . , r. ∵) Recall Uq(g) ֒→ U. v0 : h.w. vector of L(λ, φ, c) v0 : h.w. vector with weight λ as Uq(g)-module

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 14 / 25

slide-80
SLIDE 80

. . . . . .

Category Oc For c = (c1, . . . , cr) ∈ Zr, let Oc be the category of finite dim. U-modules s.t.

Ck (1 ≤ k ≤ r) acts on M ∈ Oc as the multiplication by qck. M ∈ Oc has the weight space decom.: M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K( j,l) · m = qλ( j,l)m}.

All eigenvalues of H±

(i,k),t ((i, k) ∈ Γ(m), t ≥ 0) are elements of

Z[q, q−1, (q − q−1)−1].

.

Proposition

. . .

1

simple object of Oc is a h.w. module. . .

2

If mk ≥ n for all k = 1, . . . , r − 1,

Sn,r -mod ⊂ Oc.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 15 / 25

slide-81
SLIDE 81

. . . . . .

Category Oc For c = (c1, . . . , cr) ∈ Zr, let Oc be the category of finite dim. U-modules s.t.

Ck (1 ≤ k ≤ r) acts on M ∈ Oc as the multiplication by qck. M ∈ Oc has the weight space decom.: M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K( j,l) · m = qλ( j,l)m}.

All eigenvalues of H±

(i,k),t ((i, k) ∈ Γ(m), t ≥ 0) are elements of

Z[q, q−1, (q − q−1)−1].

.

Proposition

. . .

1

simple object of Oc is a h.w. module. . .

2

If mk ≥ n for all k = 1, . . . , r − 1,

Sn,r -mod ⊂ Oc.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 15 / 25

slide-82
SLIDE 82

. . . . . .

Category Oc For c = (c1, . . . , cr) ∈ Zr, let Oc be the category of finite dim. U-modules s.t.

Ck (1 ≤ k ≤ r) acts on M ∈ Oc as the multiplication by qck. M ∈ Oc has the weight space decom.: M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K( j,l) · m = qλ( j,l)m}.

All eigenvalues of H±

(i,k),t ((i, k) ∈ Γ(m), t ≥ 0) are elements of

Z[q, q−1, (q − q−1)−1].

.

Proposition

. . .

1

simple object of Oc is a h.w. module. . .

2

If mk ≥ n for all k = 1, . . . , r − 1,

Sn,r -mod ⊂ Oc.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 15 / 25

slide-83
SLIDE 83

. . . . . .

Category Oc For c = (c1, . . . , cr) ∈ Zr, let Oc be the category of finite dim. U-modules s.t.

Ck (1 ≤ k ≤ r) acts on M ∈ Oc as the multiplication by qck. M ∈ Oc has the weight space decom.: M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K( j,l) · m = qλ( j,l)m}.

All eigenvalues of H±

(i,k),t ((i, k) ∈ Γ(m), t ≥ 0) are elements of

Z[q, q−1, (q − q−1)−1].

.

Proposition

. . .

1

simple object of Oc is a h.w. module. . .

2

If mk ≥ n for all k = 1, . . . , r − 1,

Sn,r -mod ⊂ Oc.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 15 / 25

slide-84
SLIDE 84

. . . . . .

Category Oc For c = (c1, . . . , cr) ∈ Zr, let Oc be the category of finite dim. U-modules s.t.

Ck (1 ≤ k ≤ r) acts on M ∈ Oc as the multiplication by qck. M ∈ Oc has the weight space decom.: M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K( j,l) · m = qλ( j,l)m}.

All eigenvalues of H±

(i,k),t ((i, k) ∈ Γ(m), t ≥ 0) are elements of

Z[q, q−1, (q − q−1)−1].

.

Proposition

. . .

1

simple object of Oc is a h.w. module. . .

2

If mk ≥ n for all k = 1, . . . , r − 1,

Sn,r -mod ⊂ Oc.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 15 / 25

slide-85
SLIDE 85

. . . . . .

Category Oc For c = (c1, . . . , cr) ∈ Zr, let Oc be the category of finite dim. U-modules s.t.

Ck (1 ≤ k ≤ r) acts on M ∈ Oc as the multiplication by qck. M ∈ Oc has the weight space decom.: M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K( j,l) · m = qλ( j,l)m}.

All eigenvalues of H±

(i,k),t ((i, k) ∈ Γ(m), t ≥ 0) are elements of

Z[q, q−1, (q − q−1)−1].

.

Proposition

. . .

1

simple object of Oc is a h.w. module. . .

2

If mk ≥ n for all k = 1, . . . , r − 1,

Sn,r -mod ⊂ Oc.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 15 / 25

slide-86
SLIDE 86

. . . . . .

Example

V := ⊕

(i,k)∈Γ(m) Q(q)v(i,k) with the following action:

K+

( j,l) · v(i,k) =

       qv(i,k)

if (j, l) = (i, k),

  • thewise.

H±

( j,l).t · v(i,k) =

       qtc1

if ( j, l) = (i, k),

  • therwise.

X−

( j,l),t · v(i,k) =

           qtc1 (q − q−1)t v(i+1,k)

if (j, l) = (i, k)

  • therwise

X+

( j,l),t · v(i,k)

=                    qtc1 (q − q−1)t v(i−1,k)

if i 1 and ( j, l) = (i − 1, k)

qtc1 (q − q−1)t (qc1 − qck)v(mk−1,k−1)

if i = 1 and (j, l) = (mk−1, k − 1)

  • therwise

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 16 / 25

slide-87
SLIDE 87

. . . . . .

Example

V := ⊕

(i,k)∈Γ(m) Q(q)v(i,k) with the following action:

K+

( j,l) · v(i,k) =

       qv(i,k)

if (j, l) = (i, k),

  • thewise.

H±

( j,l).t · v(i,k) =

       qtc1

if ( j, l) = (i, k),

  • therwise.

X−

( j,l),t · v(i,k) =

           qtc1 (q − q−1)t v(i+1,k)

if (j, l) = (i, k)

  • therwise

X+

( j,l),t · v(i,k)

=                    qtc1 (q − q−1)t v(i−1,k)

if i 1 and ( j, l) = (i − 1, k)

qtc1 (q − q−1)t (qc1 − qck)v(mk−1,k−1)

if i = 1 and (j, l) = (mk−1, k − 1)

  • therwise

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 16 / 25

slide-88
SLIDE 88

. . . . . .

Example

V := ⊕

(i,k)∈Γ(m) Q(q)v(i,k) with the following action:

K+

( j,l) · v(i,k) =

       qv(i,k)

if (j, l) = (i, k),

  • thewise.

H±

( j,l).t · v(i,k) =

       qtc1

if ( j, l) = (i, k),

  • therwise.

X−

( j,l),t · v(i,k) =

           qtc1 (q − q−1)t v(i+1,k)

if (j, l) = (i, k)

  • therwise

X+

( j,l),t · v(i,k)

=                    qtc1 (q − q−1)t v(i−1,k)

if i 1 and ( j, l) = (i − 1, k)

qtc1 (q − q−1)t (qc1 − qck)v(mk−1,k−1)

if i = 1 and (j, l) = (mk−1, k − 1)

  • therwise

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 16 / 25

slide-89
SLIDE 89

. . . . . .

Example

V := ⊕

(i,k)∈Γ(m) Q(q)v(i,k) with the following action:

K+

( j,l) · v(i,k) =

       qv(i,k)

if (j, l) = (i, k),

  • thewise.

H±

( j,l).t · v(i,k) =

       qtc1

if ( j, l) = (i, k),

  • therwise.

X−

( j,l),t · v(i,k) =

           qtc1 (q − q−1)t v(i+1,k)

if (j, l) = (i, k)

  • therwise

X+

( j,l),t · v(i,k)

=                    qtc1 (q − q−1)t v(i−1,k)

if i 1 and ( j, l) = (i − 1, k)

qtc1 (q − q−1)t (qc1 − qck)v(mk−1,k−1)

if i = 1 and (j, l) = (mk−1, k − 1)

  • therwise

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 16 / 25

slide-90
SLIDE 90

. . . . . .

Example

V := ⊕

(i,k)∈Γ(m) Q(q)v(i,k) with the following action:

K+

( j,l) · v(i,k) =

       qv(i,k)

if (j, l) = (i, k),

  • thewise.

H±

( j,l).t · v(i,k) =

       qtc1

if ( j, l) = (i, k),

  • therwise.

X−

( j,l),t · v(i,k) =

           qtc1 (q − q−1)t v(i+1,k)

if (j, l) = (i, k)

  • therwise

X+

( j,l),t · v(i,k)

=                    qtc1 (q − q−1)t v(i−1,k)

if i 1 and ( j, l) = (i − 1, k)

qtc1 (q − q−1)t (qc1 − qck)v(mk−1,k−1)

if i = 1 and (j, l) = (mk−1, k − 1)

  • therwise

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 16 / 25

slide-91
SLIDE 91

. . . . . .

U-module V : Q(q)v(1,1) ↓ ↑

X−

(1,1),t

X+

(1,1),t

Q(q)v(2,1) . . . Q(q)v(i−1,1) ↓ ↑

X−

(i−1,1),t

X+

(i−1,1),t

Q(q)v(i,1) ↓ ↑

X−

(i,1),t

X+

(i,1),t

Q(q)v(i+1,1) ↓ ↑

X−

(i+1,1),t

X+

(i+1,1),t

. . . Q(q)v(m1−1,1) ↓ ↑

X−

(m1−1,1),t

X+

(m1−1,1),t

Q(q)v(m1,1) ↓ ↑

X−

(m1,1),t

X+

(m1,1),t

Q(q)v(1,2) ↓ ↑

X−

(1,2),t

X+

(1,2),t

Q(q)v(2,2) . . . Q(q)v(i−1,2) ↓ ↑

X−

(i−1,2),t

X+

(i−1,2),t

Q(q)v(i,2) ↓ ↑

X−

(i,2),t

X+

(i,2),t

Q(q)v(i+1,2) ↓ ↑

X−

(i+1,2),t

X+

(i+1,2),t

. . . Q(q)v(m2−1,2) ↓ ↑

X−

(m2−1,2),t

X+

(m2−1,2),t

Q(q)v(m2,2) ↓ ↑

X−

(m2,2),t

X+

(m2,2),t

· · · Q(q)v(1,k) ↓ ↑

X−

(1,k),t

X+

(1,k),t

Q(q)v(2,k) . . . Q(q)v(i−1,k) ↓ ↑

X−

(i−1,k),t

X+

(i−1,k),t

Q(q)v(i,k) ↓ ↑

X−

(i,k),t

X+

(i,k),t

Q(q)v(i+1,k) ↓ ↑

X−

(i+1,k),t

X+

(i+1,k),t

. . . Q(q)v(mk−1,k) ↓ ↑

X−

(mk−1,k),t

X+

(mk−1,k),t

Q(q)v(mk,k) ↓ ↑

X−

(mk,k),t

X+

(mk,k),t

· · · Q(q)v(1,r) ↓ ↑

X−

(1,r),t

X+

(1,r),t

Q(q)v(2,r) . . . Q(q)v(i−1,r) ↓ ↑

X−

(i−1,r),t

X+

(i−1,r),t

Q(q)v(i,r) ↓ ↑

X−

(i,r),t

X+

(i,r),t

Q(q)v(i+1,r) ↓ ↑

X−

(i+1,r),t

X+

(i+1,r),t

. . . Q(q)v(mr−1,r) ↓ ↑

X−

(mr−1,r),t

X+

(mr−1,r),

Q(q)v(mr,r)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 17 / 25

slide-92
SLIDE 92

. . . . . .

ResU

Uq(g) V (red : omit, only t = 0) :

Q(q)v(1,1) ↓ ↑

X−

(1,1),t

X+

(1,1),t

Q(q)v(2,1) . . . Q(q)v(i−1,1) ↓ ↑

X−

(i−1,1),t

X+

(i−1,1),t

Q(q)v(i,1) ↓ ↑

X−

(i,1),t

X+

(i,1),t

Q(q)v(i+1,1) ↓ ↑

X−

(i+1,1),t

X+

(i+1,1),t

. . . Q(q)v(m1−1,1) ↓ ↑

X−

(m1−1,1),t

X+

(m1−1,1),t

Q(q)v(m1,1) ↓ ↑

X−

(m1,1),t

X+

(m1,1),t

Q(q)v(1,2) ↓ ↑

X−

(1,2),t

X+

(1,2),t

Q(q)v(2,2) . . . Q(q)v(i−1,2) ↓ ↑

X−

(i−1,2),t

X+

(i−1,2),t

Q(q)v(i,2) ↓ ↑

X−

(i,2),t

X+

(i,2),t

Q(q)v(i+1,2) ↓ ↑

X−

(i+1,2),t

X+

(i+1,2),t

. . . Q(q)v(m2−1,2) ↓ ↑

X−

(m2−1,2),t

X+

(m2−1,2),t

Q(q)v(m2,2) ↓ ↑

X−

(m2,2),t

X+

(m2,2),t

· · · Q(q)v(1,k) ↓ ↑

X−

(1,k),t

X+

(1,k),t

Q(q)v(2,k) . . . Q(q)v(i−1,k) ↓ ↑

X−

(i−1,k),t

X+

(i−1,k),t

Q(q)v(i,k) ↓ ↑

X−

(i,k),t

X+

(i,k),t

Q(q)v(i+1,k) ↓ ↑

X−

(i+1,k),t

X+

(i+1,k),t

. . . Q(q)v(mk−1,k) ↓ ↑

X−

(mk−1,k),t

X+

(mk−1,k),t

Q(q)v(mk,k) ↓ ↑

X−

(mk,k),t

X+

(mk,k),t

· · · Q(q)v(1,r) ↓ ↑

X−

(1,r),t

X+

(1,r),t

Q(q)v(2,r) . . . Q(q)v(i−1,r) ↓ ↑

X−

(i−1,r),t

X+

(i−1,r),t

Q(q)v(i,r) ↓ ↑

X−

(i,r),t

X+

(i,r),t

Q(q)v(i+1,r) ↓ ↑

X−

(i+1,r),t

X+

(i+1,r),t

. . . Q(q)v(mr−1,r) ↓ ↑

X−

(mr−1,r),t

X+

(mr−1,r),

Q(q)v(mr,r)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 17 / 25

slide-93
SLIDE 93

. . . . . .

Weyl modules and simple modules of Sn,r Assume that mk ≥ n for all k = 1, . . . , r − 1. Then we have U ↠ Sn,r.

Sn,r : quasi-hereditary algebra. Λ+

n,r(m) := {λ ∈ Λn,r(m)

  • λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) for all k = 1, . . . , r}

{W(λ)

  • λ ∈ Λ+

n,r(m)} : a set of standard (Weyl) modules of Sn,r.

{L(λ) := Top W(λ)

  • λ ∈ Λ+

n,r(m)} = { simple Sn,r-modules}/iso..

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 18 / 25

slide-94
SLIDE 94

. . . . . .

Weyl modules and simple modules of Sn,r Assume that mk ≥ n for all k = 1, . . . , r − 1. Then we have U ↠ Sn,r.

Sn,r : quasi-hereditary algebra. Λ+

n,r(m) := {λ ∈ Λn,r(m)

  • λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) for all k = 1, . . . , r}

{W(λ)

  • λ ∈ Λ+

n,r(m)} : a set of standard (Weyl) modules of Sn,r.

{L(λ) := Top W(λ)

  • λ ∈ Λ+

n,r(m)} = { simple Sn,r-modules}/iso..

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 18 / 25

slide-95
SLIDE 95

. . . . . .

Weyl modules and simple modules of Sn,r Assume that mk ≥ n for all k = 1, . . . , r − 1. Then we have U ↠ Sn,r.

Sn,r : quasi-hereditary algebra. Λ+

n,r(m) := {λ ∈ Λn,r(m)

  • λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) for all k = 1, . . . , r}

{W(λ)

  • λ ∈ Λ+

n,r(m)} : a set of standard (Weyl) modules of Sn,r.

{L(λ) := Top W(λ)

  • λ ∈ Λ+

n,r(m)} = { simple Sn,r-modules}/iso..

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 18 / 25

slide-96
SLIDE 96

. . . . . .

Weyl modules and simple modules of Sn,r Assume that mk ≥ n for all k = 1, . . . , r − 1. Then we have U ↠ Sn,r.

Sn,r : quasi-hereditary algebra. Λ+

n,r(m) := {λ ∈ Λn,r(m)

  • λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) for all k = 1, . . . , r}

{W(λ)

  • λ ∈ Λ+

n,r(m)} : a set of standard (Weyl) modules of Sn,r.

{L(λ) := Top W(λ)

  • λ ∈ Λ+

n,r(m)} = { simple Sn,r-modules}/iso..

.

Theorem (W)

. . Under Uq(g) ֒→ U,

W(λ) ⊕

µ∈Λ+

n,r(m)

( W(µ[1])⊗W(µ[2])⊗· · ·⊗W(µ[r]) )⊕βλµ as Uq(g)-modules, βλµ is computed by a generalization of Littlewood-Richardson rule.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 18 / 25

slide-97
SLIDE 97

. . . . . .

Weyl modules and simple modules of Sn,r Assume that mk ≥ n for all k = 1, . . . , r − 1. Then we have U ↠ Sn,r.

Sn,r : quasi-hereditary algebra. Λ+

n,r(m) := {λ ∈ Λn,r(m)

  • λ(1,k) ≥ λ(2,k) ≥ · · · ≥ λ(mk,k) for all k = 1, . . . , r}

{W(λ)

  • λ ∈ Λ+

n,r(m)} : a set of standard (Weyl) modules of Sn,r.

{L(λ) := Top W(λ)

  • λ ∈ Λ+

n,r(m)} = { simple Sn,r-modules}/iso..

.

Proposition

. . As a U-module,

W(λ) (resp. L(λ)) is a h.w. module with h.w. (λ, φ, c), where φ±

(i,k),t =

±1 q±(1−t)(q − q−1)Φ±

t (qck+2(1−i), qck+2(2−i), . . . , qck+2(λ(i,k)−i))

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 18 / 25

slide-98
SLIDE 98

. . . . . .

Harish-Chandra Ind and Res Recall the generators of U :

X = { X±

(i,k),t, K± ( j,l), H± ( j,l),t, Ck

  • (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r

} .

Put

UP : subalg. of U gen. by X \ {X−

(mk,k),t

  • 1 ≤ k ≤ r − 1, t ≥ 0}

UL : subalg. of U gen. by X \ {X±

(mk,k),t

  • 1 ≤ k ≤ r − 1, t ≥ 0}

.

Lemma

. .

UL ֒→ UP ֒→ U

− →

↠

id

g UL U[1] ⊗ U[2] ⊗ · · · ⊗ U[r]

where U[k] is an ass. algebra generated by

{ X±

(i,k),t, K± ( j,k), H± ( j,k),t,Ck

  • 1 ≤ i ≤ mk − 1, 1 ≤ j ≤ mk, t ≥ 0

}

with the same defining relations of U.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 19 / 25

slide-99
SLIDE 99

. . . . . .

Harish-Chandra Ind and Res Recall the generators of U :

X = { X±

(i,k),t, K± ( j,l), H± ( j,l),t, Ck

  • (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r

} .

Put

UP : subalg. of U gen. by X \ {X−

(mk,k),t

  • 1 ≤ k ≤ r − 1, t ≥ 0}

UL : subalg. of U gen. by X \ {X±

(mk,k),t

  • 1 ≤ k ≤ r − 1, t ≥ 0}

.

Lemma

. .

UL ֒→ UP ֒→ U

− →

↠

id

g UL U[1] ⊗ U[2] ⊗ · · · ⊗ U[r]

where U[k] is an ass. algebra generated by

{ X±

(i,k),t, K± ( j,k), H± ( j,k),t,Ck

  • 1 ≤ i ≤ mk − 1, 1 ≤ j ≤ mk, t ≥ 0

}

with the same defining relations of U.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 19 / 25

slide-100
SLIDE 100

. . . . . .

Harish-Chandra Ind and Res Recall the generators of U :

X = { X±

(i,k),t, K± ( j,l), H± ( j,l),t, Ck

  • (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r

} .

Put

UP : subalg. of U gen. by X \ {X−

(mk,k),t

  • 1 ≤ k ≤ r − 1, t ≥ 0}

UL : subalg. of U gen. by X \ {X±

(mk,k),t

  • 1 ≤ k ≤ r − 1, t ≥ 0}

.

Lemma

. .

UL ֒→ UP ֒→ U

− →

↠

id

g UL U[1] ⊗ U[2] ⊗ · · · ⊗ U[r]

where U[k] is an ass. algebra generated by

{ X±

(i,k),t, K± ( j,k), H± ( j,k),t,Ck

  • 1 ≤ i ≤ mk − 1, 1 ≤ j ≤ mk, t ≥ 0

}

with the same defining relations of U.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 19 / 25

slide-101
SLIDE 101

. . . . . .

Harish-Chandra Ind and Res Recall the generators of U :

X = { X±

(i,k),t, K± ( j,l), H± ( j,l),t, Ck

  • (i, k) ∈ Γ′(m), (j, l) ∈ Γ(m), t ≥ 0, 1 ≤ k ≤ r

} .

Put

UP : subalg. of U gen. by X \ {X−

(mk,k),t

  • 1 ≤ k ≤ r − 1, t ≥ 0}

UL : subalg. of U gen. by X \ {X±

(mk,k),t

  • 1 ≤ k ≤ r − 1, t ≥ 0}

.

Lemma

. .

UL ֒→ UP ֒→ U

− →

↠

id

g UL U[1] ⊗ U[2] ⊗ · · · ⊗ U[r]

where U[k] is an ass. algebra generated by

{ X±

(i,k),t, K± ( j,k), H± ( j,k),t,Ck

  • 1 ≤ i ≤ mk − 1, 1 ≤ j ≤ mk, t ≥ 0

}

with the same defining relations of U.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 19 / 25

slide-102
SLIDE 102

. . . . . .

U-module V : Q(q)v(1,1) ↓ ↑

X−

(1,1),t

X+

(1,1),t

Q(q)v(2,1) . . . Q(q)v(i−1,1) ↓ ↑

X−

(i−1,1),t

X+

(i−1,1),t

Q(q)v(i,1) ↓ ↑

X−

(i,1),t

X+

(i,1),t

Q(q)v(i+1,1) ↓ ↑

X−

(i+1,1),t

X+

(i+1,1),t

. . . Q(q)v(m1−1,1) ↓ ↑

X−

(m1−1,1),t

X+

(m1−1,1),t

Q(q)v(m1,1) ↓ ↑

X−

(m1,1),t

X+

(m1,1),t

Q(q)v(1,2) ↓ ↑

X−

(1,2),t

X+

(1,2),t

Q(q)v(2,2) . . . Q(q)v(i−1,2) ↓ ↑

X−

(i−1,2),t

X+

(i−1,2),t

Q(q)v(i,2) ↓ ↑

X−

(i,2),t

X+

(i,2),t

Q(q)v(i+1,2) ↓ ↑

X−

(i+1,2),t

X+

(i+1,2),t

. . . Q(q)v(m2−1,2) ↓ ↑

X−

(m2−1,2),t

X+

(m2−1,2),t

Q(q)v(m2,2) ↓ ↑

X−

(m2,2),t

X+

(m2,2),t

· · · Q(q)v(1,k) ↓ ↑

X−

(1,k),t

X+

(1,k),t

Q(q)v(2,k) . . . Q(q)v(i−1,k) ↓ ↑

X−

(i−1,k),t

X+

(i−1,k),t

Q(q)v(i,k) ↓ ↑

X−

(i,k),t

X+

(i,k),t

Q(q)v(i+1,k) ↓ ↑

X−

(i+1,k),t

X+

(i+1,k),t

. . . Q(q)v(mk−1,k) ↓ ↑

X−

(mk−1,k),t

X+

(mk−1,k),t

Q(q)v(mk,k) ↓ ↑

X−

(mk,k),t

X+

(mk,k),t

· · · Q(q)v(1,r) ↓ ↑

X−

(1,r),t

X+

(1,r),t

Q(q)v(2,r) . . . Q(q)v(i−1,r) ↓ ↑

X−

(i−1,r),t

X+

(i−1,r),t

Q(q)v(i,r) ↓ ↑

X−

(i,r),t

X+

(i,r),t

Q(q)v(i+1,r) ↓ ↑

X−

(i+1,r),t

X+

(i+1,r),t

. . . Q(q)v(mr−1,r) ↓ ↑

X−

(mr−1,r),t

X+

(mr−1,r),

Q(q)v(mr,r)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 20 / 25

slide-103
SLIDE 103

. . . . . .

ResU

UP V (red : omit) :

Q(q)v(1,1) ↓ ↑

X−

(1,1),t

X+

(1,1),t

Q(q)v(2,1) . . . Q(q)v(i−1,1) ↓ ↑

X−

(i−1,1),t

X+

(i−1,1),t

Q(q)v(i,1) ↓ ↑

X−

(i,1),t

X+

(i,1),t

Q(q)v(i+1,1) ↓ ↑

X−

(i+1,1),t

X+

(i+1,1),t

. . . Q(q)v(m1−1,1) ↓ ↑

X−

(m1−1,1),t

X+

(m1−1,1),t

Q(q)v(m1,1) ↓ ↑

X−

(m1,1),t

X+

(m1,1),t

Q(q)v(1,2) ↓ ↑

X−

(1,2),t

X+

(1,2),t

Q(q)v(2,2) . . . Q(q)v(i−1,2) ↓ ↑

X−

(i−1,2),t

X+

(i−1,2),t

Q(q)v(i,2) ↓ ↑

X−

(i,2),t

X+

(i,2),t

Q(q)v(i+1,2) ↓ ↑

X−

(i+1,2),t

X+

(i+1,2),t

. . . Q(q)v(m2−1,2) ↓ ↑

X−

(m2−1,2),t

X+

(m2−1,2),t

Q(q)v(m2,2) ↓ ↑

X−

(m2,2),t

X+

(m2,2),t

· · · Q(q)v(1,k) ↓ ↑

X−

(1,k),t

X+

(1,k),t

Q(q)v(2,k) . . . Q(q)v(i−1,k) ↓ ↑

X−

(i−1,k),t

X+

(i−1,k),t

Q(q)v(i,k) ↓ ↑

X−

(i,k),t

X+

(i,k),t

Q(q)v(i+1,k) ↓ ↑

X−

(i+1,k),t

X+

(i+1,k),t

. . . Q(q)v(mk−1,k) ↓ ↑

X−

(mk−1,k),t

X+

(mk−1,k),t

Q(q)v(mk,k) ↓ ↑

X−

(mk,k),t

X+

(mk,k),t

· · · Q(q)v(1,r) ↓ ↑

X−

(1,r),t

X+

(1,r),t

Q(q)v(2,r) . . . Q(q)v(i−1,r) ↓ ↑

X−

(i−1,r),t

X+

(i−1,r),t

Q(q)v(i,r) ↓ ↑

X−

(i,r),t

X+

(i,r),t

Q(q)v(i+1,r) ↓ ↑

X−

(i+1,r),t

X+

(i+1,r),t

. . . Q(q)v(mr−1,r) ↓ ↑

X−

(mr−1,r),t

X+

(mr−1,r),

Q(q)v(mr,r)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 20 / 25

slide-104
SLIDE 104

. . . . . .

ResU

UL V (red : omit) :

Q(q)v(1,1) ↓ ↑

X−

(1,1),t

X+

(1,1),t

Q(q)v(2,1) . . . Q(q)v(i−1,1) ↓ ↑

X−

(i−1,1),t

X+

(i−1,1),t

Q(q)v(i,1) ↓ ↑

X−

(i,1),t

X+

(i,1),t

Q(q)v(i+1,1) ↓ ↑

X−

(i+1,1),t

X+

(i+1,1),t

. . . Q(q)v(m1−1,1) ↓ ↑

X−

(m1−1,1),t

X+

(m1−1,1),t

Q(q)v(m1,1) ↓ ↑

X−

(m1,1),t

X+

(m1,1),t

Q(q)v(1,2) ↓ ↑

X−

(1,2),t

X+

(1,2),t

Q(q)v(2,2) . . . Q(q)v(i−1,2) ↓ ↑

X−

(i−1,2),t

X+

(i−1,2),t

Q(q)v(i,2) ↓ ↑

X−

(i,2),t

X+

(i,2),t

Q(q)v(i+1,2) ↓ ↑

X−

(i+1,2),t

X+

(i+1,2),t

. . . Q(q)v(m2−1,2) ↓ ↑

X−

(m2−1,2),t

X+

(m2−1,2),t

Q(q)v(m2,2) ↓ ↑

X−

(m2,2),t

X+

(m2,2),t

· · · Q(q)v(1,k) ↓ ↑

X−

(1,k),t

X+

(1,k),t

Q(q)v(2,k) . . . Q(q)v(i−1,k) ↓ ↑

X−

(i−1,k),t

X+

(i−1,k),t

Q(q)v(i,k) ↓ ↑

X−

(i,k),t

X+

(i,k),t

Q(q)v(i+1,k) ↓ ↑

X−

(i+1,k),t

X+

(i+1,k),t

. . . Q(q)v(mk−1,k) ↓ ↑

X−

(mk−1,k),t

X+

(mk−1,k),t

Q(q)v(mk,k) ↓ ↑

X−

(mk,k),t

X+

(mk,k),t

· · · Q(q)v(1,r) ↓ ↑

X−

(1,r),t

X+

(1,r),t

Q(q)v(2,r) . . . Q(q)v(i−1,r) ↓ ↑

X−

(i−1,r),t

X+

(i−1,r),t

Q(q)v(i,r) ↓ ↑

X−

(i,r),t

X+

(i,r),t

Q(q)v(i+1,r) ↓ ↑

X−

(i+1,r),t

X+

(i+1,r),t

. . . Q(q)v(mr−1,r) ↓ ↑

X−

(mr−1,r),t

X+

(mr−1,r),

Q(q)v(mr,r)

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 20 / 25

slide-105
SLIDE 105

. . . . . .

Harish-Chandra Ind and Res Define

RU

UL : UL -mod → UP -mod → U -mod.

∈ ∈ ∈ N → N (via g) → U ⊗UP N.

∗RU UL : U -mod → UP -mod → UL -mod.

∈ ∈ ∈ M → ResU

UP M → {m ∈ M

  • Ker g · m = 0}.

RU

UL is left adjoint to ∗RU UL,

i.e. HomU

(RU

UL(N), M) HomUL (N, ∗RU UL(M)).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 21 / 25

slide-106
SLIDE 106

. . . . . .

Harish-Chandra Ind and Res Define

RU

UL : UL -mod → UP -mod → U -mod.

∈ ∈ ∈ N → N (via g) → U ⊗UP N.

∗RU UL : U -mod → UP -mod → UL -mod.

∈ ∈ ∈ M → ResU

UP M → {m ∈ M

  • Ker g · m = 0}.

RU

UL is left adjoint to ∗RU UL,

i.e. HomU

(RU

UL(N), M) HomUL (N, ∗RU UL(M)).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 21 / 25

slide-107
SLIDE 107

. . . . . .

Harish-Chandra Ind and Res Define

RU

UL : UL -mod → UP -mod → U -mod.

∈ ∈ ∈ N → N (via g) → U ⊗UP N.

∗RU UL : U -mod → UP -mod → UL -mod.

∈ ∈ ∈ M → ResU

UP M → {m ∈ M

  • Ker g · m = 0}.

RU

UL is left adjoint to ∗RU UL,

i.e. HomU

(RU

UL(N), M) HomUL (N, ∗RU UL(M)).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 21 / 25

slide-108
SLIDE 108

. . . . . .

Harish-Chandra Ind and Res Recall U0 ֒→ UL ֒→ UP ֒→ U. .

Proposition

. .

N(λ,φ,c) : h.w. UL-module with h.w. (λ, φ, c) RU

ULN(λ,φ,c) : h.w. U-module with h.w. (λ, φ, c).

In particular,

Top RU

ULN(λ,φ,c) L(λ, φ, c).

Assume that L(λ, φ, c) is fin. dim.

∗RU

ULL(λ, φ, c) : simple h.w. UL-module with h.w. (λ, φ, c).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 22 / 25

slide-109
SLIDE 109

. . . . . .

Harish-Chandra Ind and Res Recall U0 ֒→ UL ֒→ UP ֒→ U. .

Proposition

. .

N(λ,φ,c) : h.w. UL-module with h.w. (λ, φ, c) RU

ULN(λ,φ,c) : h.w. U-module with h.w. (λ, φ, c).

In particular,

Top RU

ULN(λ,φ,c) L(λ, φ, c).

Assume that L(λ, φ, c) is fin. dim.

∗RU

ULL(λ, φ, c) : simple h.w. UL-module with h.w. (λ, φ, c).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 22 / 25

slide-110
SLIDE 110

. . . . . .

Harish-Chandra Ind and Res Recall U0 ֒→ UL ֒→ UP ֒→ U. .

Proposition

. .

N(λ,φ,c) : h.w. UL-module with h.w. (λ, φ, c) RU

ULN(λ,φ,c) : h.w. U-module with h.w. (λ, φ, c).

In particular,

Top RU

ULN(λ,φ,c) L(λ, φ, c).

Assume that L(λ, φ, c) is fin. dim.

∗RU

ULL(λ, φ, c) : simple h.w. UL-module with h.w. (λ, φ, c).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 22 / 25

slide-111
SLIDE 111

. . . . . .

Harish-Chandra Ind and Res Recall U0 ֒→ UL ֒→ UP ֒→ U. .

Proposition

. .

N(λ,φ,c) : h.w. UL-module with h.w. (λ, φ, c) RU

ULN(λ,φ,c) : h.w. U-module with h.w. (λ, φ, c).

In particular,

Top RU

ULN(λ,φ,c) L(λ, φ, c).

Assume that L(λ, φ, c) is fin. dim.

∗RU

ULL(λ, φ, c) : simple h.w. UL-module with h.w. (λ, φ, c).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 22 / 25

slide-112
SLIDE 112

. . . . . .

Harish-Chandra Ind and Res Recall U0 ֒→ UL ֒→ UP ֒→ U. .

Proposition

. .

N(λ,φ,c) : h.w. UL-module with h.w. (λ, φ, c) RU

ULN(λ,φ,c) : h.w. U-module with h.w. (λ, φ, c).

In particular,

Top RU

ULN(λ,φ,c) L(λ, φ, c).

Assume that L(λ, φ, c) is fin. dim.

∗RU

ULL(λ, φ, c) : simple h.w. UL-module with h.w. (λ, φ, c).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 22 / 25

slide-113
SLIDE 113

. . . . . .

Harish-Chandra Ind and Res Recall U0 ֒→ UL ֒→ UP ֒→ U. .

Proposition

. .

N(λ,φ,c) : h.w. UL-module with h.w. (λ, φ, c) RU

ULN(λ,φ,c) : h.w. U-module with h.w. (λ, φ, c).

In particular,

Top RU

ULN(λ,φ,c) L(λ, φ, c).

Assume that L(λ, φ, c) is fin. dim.

∗RU

ULL(λ, φ, c) : simple h.w. UL-module with h.w. (λ, φ, c).

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 22 / 25

slide-114
SLIDE 114

. . . . . .

evaluation functors Recall g = glm1 ⊕ · · · ⊕ glmr ⊂ glm.

Og : category of finite dim. Uq(g)-modules

s.t. M ∈ Og has the weight space decom.:

M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K(i,k) · m = qλ(i,k)m}.

We have

Og = ⊕

(n1,...,nr)∈Zr

≥0

Sn1,1 ⊗ Sn2,1 ⊗ · · · ⊗ Snr,1 -mod

For each c = (c1, . . . , cr) ∈ Zr and (n1, . . . , nr) ∈ Zr

≥0, we have

UL U[1] ⊗ · · · ⊗ U[r] ↠ Sn1,1 ⊗ · · · ⊗ Snr,1

(Note the relation (T0 − qck) = 0 in Hnk,1) Through this surjection, we have

evc : Og → UL -mod

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 23 / 25

slide-115
SLIDE 115

. . . . . .

evaluation functors Recall g = glm1 ⊕ · · · ⊕ glmr ⊂ glm.

Og : category of finite dim. Uq(g)-modules

s.t. M ∈ Og has the weight space decom.:

M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K(i,k) · m = qλ(i,k)m}.

We have

Og = ⊕

(n1,...,nr)∈Zr

≥0

Sn1,1 ⊗ Sn2,1 ⊗ · · · ⊗ Snr,1 -mod

For each c = (c1, . . . , cr) ∈ Zr and (n1, . . . , nr) ∈ Zr

≥0, we have

UL U[1] ⊗ · · · ⊗ U[r] ↠ Sn1,1 ⊗ · · · ⊗ Snr,1

(Note the relation (T0 − qck) = 0 in Hnk,1) Through this surjection, we have

evc : Og → UL -mod

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 23 / 25

slide-116
SLIDE 116

. . . . . .

evaluation functors Recall g = glm1 ⊕ · · · ⊕ glmr ⊂ glm.

Og : category of finite dim. Uq(g)-modules

s.t. M ∈ Og has the weight space decom.:

M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K(i,k) · m = qλ(i,k)m}.

We have

Og = ⊕

(n1,...,nr)∈Zr

≥0

Sn1,1 ⊗ Sn2,1 ⊗ · · · ⊗ Snr,1 -mod

For each c = (c1, . . . , cr) ∈ Zr and (n1, . . . , nr) ∈ Zr

≥0, we have

UL U[1] ⊗ · · · ⊗ U[r] ↠ Sn1,1 ⊗ · · · ⊗ Snr,1

(Note the relation (T0 − qck) = 0 in Hnk,1) Through this surjection, we have

evc : Og → UL -mod

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 23 / 25

slide-117
SLIDE 117

. . . . . .

evaluation functors Recall g = glm1 ⊕ · · · ⊕ glmr ⊂ glm.

Og : category of finite dim. Uq(g)-modules

s.t. M ∈ Og has the weight space decom.:

M = ⊕

λ∈P≥0

Mλ, where Mλ = {m ∈ M

  • K(i,k) · m = qλ(i,k)m}.

We have

Og = ⊕

(n1,...,nr)∈Zr

≥0

Sn1,1 ⊗ Sn2,1 ⊗ · · · ⊗ Snr,1 -mod

For each c = (c1, . . . , cr) ∈ Zr and (n1, . . . , nr) ∈ Zr

≥0, we have

UL U[1] ⊗ · · · ⊗ U[r] ↠ Sn1,1 ⊗ · · · ⊗ Snr,1

(Note the relation (T0 − qck) = 0 in Hnk,1) Through this surjection, we have

evc : Og → UL -mod

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 23 / 25

slide-118
SLIDE 118

. . . . . .

Og

evc

−→ UL -mod

RU

UL

−→ U -mod. U -mod

∗RU UL

−→ UL -mod

ResUL

Uq(g)

−→ Uq(g) -mod

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. For λ ∈ Λ+

n,r(m), let

L(λ) : simple Sn,r-module with h.w. (λ, φ, c). L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ∈ Og : simple Uq(g)-module with h.w. λ.

We have . .

1

L(λ) Top ( RU

UL ◦ evc

(L(λ[1]) ⊗ · · · ⊗ L(λ[r])))

as U-modules. . .

2

L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ResUL

Uq(g) ◦ ∗RU UL

(L(λ)) as Uq(g)-modules.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 24 / 25

slide-119
SLIDE 119

. . . . . .

Og

evc

−→ UL -mod

RU

UL

−→ U -mod. U -mod

∗RU UL

−→ UL -mod

ResUL

Uq(g)

−→ Uq(g) -mod

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. For λ ∈ Λ+

n,r(m), let

L(λ) : simple Sn,r-module with h.w. (λ, φ, c). L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ∈ Og : simple Uq(g)-module with h.w. λ.

We have . .

1

L(λ) Top ( RU

UL ◦ evc

(L(λ[1]) ⊗ · · · ⊗ L(λ[r])))

as U-modules. . .

2

L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ResUL

Uq(g) ◦ ∗RU UL

(L(λ)) as Uq(g)-modules.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 24 / 25

slide-120
SLIDE 120

. . . . . .

Og

evc

−→ UL -mod

RU

UL

−→ U -mod. U -mod

∗RU UL

−→ UL -mod

ResUL

Uq(g)

−→ Uq(g) -mod

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. For λ ∈ Λ+

n,r(m), let

L(λ) : simple Sn,r-module with h.w. (λ, φ, c). L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ∈ Og : simple Uq(g)-module with h.w. λ.

We have . .

1

L(λ) Top ( RU

UL ◦ evc

(L(λ[1]) ⊗ · · · ⊗ L(λ[r])))

as U-modules. . .

2

L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ResUL

Uq(g) ◦ ∗RU UL

(L(λ)) as Uq(g)-modules.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 24 / 25

slide-121
SLIDE 121

. . . . . .

Og

evc

−→ UL -mod

RU

UL

−→ U -mod. U -mod

∗RU UL

−→ UL -mod

ResUL

Uq(g)

−→ Uq(g) -mod

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. For λ ∈ Λ+

n,r(m), let

L(λ) : simple Sn,r-module with h.w. (λ, φ, c). L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ∈ Og : simple Uq(g)-module with h.w. λ.

We have . .

1

L(λ) Top ( RU

UL ◦ evc

(L(λ[1]) ⊗ · · · ⊗ L(λ[r])))

as U-modules. . .

2

L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ResUL

Uq(g) ◦ ∗RU UL

(L(λ)) as Uq(g)-modules.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 24 / 25

slide-122
SLIDE 122

. . . . . .

Og

evc

−→ UL -mod

RU

UL

−→ U -mod. U -mod

∗RU UL

−→ UL -mod

ResUL

Uq(g)

−→ Uq(g) -mod

.

Theorem (W)

. . Assume that mk ≥ n for all k = 1, . . . , r − 1. For λ ∈ Λ+

n,r(m), let

L(λ) : simple Sn,r-module with h.w. (λ, φ, c). L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ∈ Og : simple Uq(g)-module with h.w. λ.

We have . .

1

L(λ) Top ( RU

UL ◦ evc

(L(λ[1]) ⊗ · · · ⊗ L(λ[r])))

as U-modules. . .

2

L(λ[1]) ⊗ · · · ⊗ L(λ[r]) ResUL

Uq(g) ◦ ∗RU UL

(L(λ)) as Uq(g)-modules.

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 24 / 25

slide-123
SLIDE 123

. . . . . .

Further problems More study for representation theory of U. Related combinatorics. Connection with (affine) Lie algebra, W-algebra, . . . . A structure on Oc as a tensor category ? (U : Hopf algebra ?) Geometric interpretation of U and its representations ?

. . .

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 25 / 25

slide-124
SLIDE 124

. . . . . .

Further problems More study for representation theory of U. Related combinatorics. Connection with (affine) Lie algebra, W-algebra, . . . . A structure on Oc as a tensor category ? (U : Hopf algebra ?) Geometric interpretation of U and its representations ?

. . .

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 25 / 25

slide-125
SLIDE 125

. . . . . .

Further problems More study for representation theory of U. Related combinatorics. Connection with (affine) Lie algebra, W-algebra, . . . . A structure on Oc as a tensor category ? (U : Hopf algebra ?) Geometric interpretation of U and its representations ?

. . .

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 25 / 25

slide-126
SLIDE 126

. . . . . .

Further problems More study for representation theory of U. Related combinatorics. Connection with (affine) Lie algebra, W-algebra, . . . . A structure on Oc as a tensor category ? (U : Hopf algebra ?) Geometric interpretation of U and its representations ?

. . .

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 25 / 25

slide-127
SLIDE 127

. . . . . .

Further problems More study for representation theory of U. Related combinatorics. Connection with (affine) Lie algebra, W-algebra, . . . . A structure on Oc as a tensor category ? (U : Hopf algebra ?) Geometric interpretation of U and its representations ?

. . .

Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q-Schur algebras 12th March, 2012 25 / 25