Cellular structures on Hecke algebras of type B

Cellular structures on Hecke algebras of type B

C´edric Bonnaf´e

CNRS (UMR 6623) - Universit´e de Franche-Comt´e (Besan¸con)

Sydney, June 2007

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 1 / 25

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 2 / 25

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 2 / 25

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

Joint work with Nicolas Jacon (Besan¸con, France)

Cellular structure

s

Canonical basis of Fock space: Ariki’s Theorem

Kazhdan-Lusztig theory and cellular structures: Geck’s Theorem

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 3 / 25

Joint work with Nicolas Jacon (Besan¸con, France) Cellular structure

s

Canonical basis of Fock space: Ariki’s Theorem

Kazhdan-Lusztig theory and cellular structures: Geck’s Theorem

Joint work with Nicolas Jacon (Besan¸con, France) Cellular structure

s

Canonical basis of Fock space: Ariki’s Theorem

Kazhdan-Lusztig theory and cellular structures: Geck’s Theorem

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 3 / 25

Joint work with Nicolas Jacon (Besan¸con, France) Cellular structure

s

Canonical basis of Fock space: Ariki’s Theorem

Kazhdan-Lusztig theory and cellular structures: Geck’s Theorem

The set-up

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 4 / 25

The set-up Weyl group, Hecke algebra

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q.

The set-up Weyl group, Hecke algebra

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 5 / 25

The set-up Weyl group, Hecke algebra

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function

R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q.

The set-up Weyl group, Hecke algebra

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 5 / 25

The set-up Simple modules, decomposition map

K =Frac(R),KH_n=K ⊗_R H_n split semisimple

IrrKH_n={V_λ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q0,q0 ∈C^×, specialization −→CHn =C⊗R Hn

R0(CHn) := Grothendieck group of CHn 'ZIrrCHn

decomposition mapd_n:R₀(KH_n)−→R₀(CH_n)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

The set-up Simple modules, decomposition map

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKH_n={V_λ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q0,q0 ∈C^×, specialization −→CHn =C⊗R Hn

R0(CHn) := Grothendieck group of CHn 'ZIrrCHn

decomposition mapd_n:R₀(KH_n)−→R₀(CH_n)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 6 / 25

The set-up Simple modules, decomposition map

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKH_n={V_λ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q0,q0 ∈C^×, specialization −→CHn =C⊗R Hn

R0(CHn) := Grothendieck group of CHn 'ZIrrCHn

decomposition mapd_n:R₀(KH_n)−→R₀(CH_n)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

The set-up Simple modules, decomposition map

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKH_n={V_λ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q0,q0 ∈C^×, specialization −→CHn =C⊗R Hn

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n

decomposition mapd_n:R₀(KH_n)−→R₀(CH_n)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 6 / 25

The set-up Simple modules, decomposition map

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKH_n={V_λ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q0,q0 ∈C^×, specialization −→CHn =C⊗R Hn

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n decomposition mapd_n:R₀(KH_n)−→R₀(CH_n)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

The set-up Simple modules, decomposition map

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKH_n={V_λ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q0,q0 ∈C^×, specialization −→CHn =C⊗R Hn

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n decomposition mapd_n:R₀(KH_n)−→R₀(CH_n)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 6 / 25

Ariki’s Theorem

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

Ariki’s Theorem Fock space

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:F_r →F_r and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)λ∈Bip is called thestandard basis

(G(λ,r))_λ∈Bip is called theKashiwara-Lusztig canonical basis

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 8 / 25

Ariki’s Theorem Fock space

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate

Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:F_r →F_r and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)λ∈Bip is called thestandard basis

(G(λ,r))_λ∈Bip is called theKashiwara-Lusztig canonical basis

Ariki’s Theorem Fock space

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:F_r →F_r and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)λ∈Bip is called thestandard basis

(G(λ,r))_λ∈Bip is called theKashiwara-Lusztig canonical basis

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 8 / 25

Ariki’s Theorem Fock space

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri F_r is endowed with an action of U_v( ^sl_e)

depending on r Uglovhas constructed an involution¯:F_r →F_r and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)λ∈Bip is called thestandard basis

(G(λ,r))_λ∈Bip is called theKashiwara-Lusztig canonical basis

Ariki’s Theorem Fock space

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r

Uglovhas constructed an involution¯:F_r →F_r and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)λ∈Bip is called thestandard basis

(G(λ,r))_λ∈Bip is called theKashiwara-Lusztig canonical basis

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 8 / 25

Ariki’s Theorem Fock space

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:F_r →F_r and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v]

Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)λ∈Bip is called thestandard basis

(G(λ,r))_λ∈Bip is called theKashiwara-Lusztig canonical basis

Ariki’s Theorem Fock space

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:F_r →F_r and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v)|λ,ri (note thatd_λλ^r (v) =1).

(|λ,ri)λ∈Bip is called thestandard basis

(G(λ,r))_λ∈Bip is called theKashiwara-Lusztig canonical basis

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 8 / 25

Ariki’s Theorem Fock space

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:F_r →F_r and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v)|λ,ri (note thatd_λλ^r (v) =1).

(|λ,ri)λ∈Bip is called the standard basis

(G(λ,r))_λ∈Bip is called the Kashiwara-Lusztig canonical basis

Ariki’s Theorem Ariki’s Theorem

Ariki’s Theorem (Ariki, Uglov, Geck-Jacon). Assume that r ≡d mod e. There exists a subset Bip_e,r(n) of Bip(n) and a bijection

Bip_e,r(n) −→ IrrCH_n

λ 7−→ D_λ^e,r

such that

U_v( ^sl_e) |∅,ri= ⊕

λ∈Bip_e,rC(v) G(λ,r), where Bip_e,r = a

n>0

Bip_e,r(n)

If λ∈Bip(n), then d_n[V_λ] = X

µ∈Bip_e,r(n)

d_λµ^r (1) [D_µ^e,r]

Remark - d_λµ^r (v) is “computable”

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 9 / 25

Ariki’s Theorem Ariki’s Theorem

Ariki’s Theorem (Ariki, Uglov, Geck-Jacon). Assume that r ≡d mod e. There exists a subset Bip_e,r(n) of Bip(n) and a bijection

Bip_e,r(n) −→ IrrCH_n

λ 7−→ D_λ^e,r

such that

U_v( ^sl_e) |∅,ri= ⊕

λ∈Bip_e,rC(v) G(λ,r), where Bip_e,r = a

n>0

Bip_e,r(n)

If λ∈Bip(n), then d_n[V_λ] = X

µ∈Bip_e,r(n)

d_λµ^r (1) [D_µ^e,r]

Remark - d_λµ^r (v) is “computable”

Ariki’s Theorem Ariki’s Theorem

Ariki’s Theorem (Ariki, Uglov, Geck-Jacon). Assume that r ≡d mod e. There exists a subset Bip_e,r(n) of Bip(n) and a bijection

Bip_e,r(n) −→ IrrCH_n

λ 7−→ D_λ^e,r

such that

U_v( ^sl_e) |∅,ri= ⊕

λ∈Bip_e,rC(v) G(λ,r), where Bip_e,r = a

n>0

Bip_e,r(n)

If λ∈Bip(n), then d_n[V_λ] = X

µ∈Bip_e,r(n)

d_λµ^r (1) [D_µ^e,r]

Remark - d_λµ^r (v) is “computable”

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 9 / 25

Ariki’s Theorem Ariki’s Theorem

Ariki’s Theorem (Ariki, Uglov, Geck-Jacon). Assume that r ≡d mod e. There exists a subset Bip_e,r(n) of Bip(n) and a bijection

Bip_e,r(n) −→ IrrCH_n

λ 7−→ D_λ^e,r

such that

U_v( ^sl_e) |∅,ri= ⊕

λ∈Bip_e,rC(v) G(λ,r), where Bip_e,r = a

n>0

Bip_e,r(n)

If λ∈Bip(n), then d_n[V_λ] = X

µ∈Bip_e,r(n)

d_λµ^r (1) [D_µ^e,r]

Remark - d_λµ^r (v) is “computable”

Ariki’s Theorem Ariki’s Theorem

Comments:

|Bip_e,r(n)|=|Bip_e,r+ke(n)| if k ∈Z (for a bijection, see Jacon, Jacon-Lecouvey)

If r >n−1,Bip_e,r(n) ={Kleshchev bipartitions} (see Ariki: it is related to the Dipper-James-Mathas or to the Graham-Lehrer cellular structure).

Bip_d0,e(n) ={FLOTW bipartitions} (Jacon). Here, d₀ ≡d mod e and d₀ ∈{0,1,2, . . . ,e−1}.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 10 / 25

Ariki’s Theorem Ariki’s Theorem

Comments:

|Bip_e,r(n)|=|Bip_e,r+ke(n)| if k ∈Z (for a bijection, see Jacon, Jacon-Lecouvey)

If r >n−1,Bip_e,r(n) ={Kleshchev bipartitions} (see Ariki: it is related to the Dipper-James-Mathas or to the Graham-Lehrer cellular structure).

Bip_d0,e(n) ={FLOTW bipartitions} (Jacon). Here, d₀ ≡d mod e and d₀ ∈{0,1,2, . . . ,e−1}.

Ariki’s Theorem Ariki’s Theorem

Comments:

|Bip_e,r(n)|=|Bip_e,r+ke(n)| if k ∈Z (for a bijection, see Jacon, Jacon-Lecouvey)

If r >n−1,Bip_e,r(n) ={Kleshchev bipartitions} (see Ariki: it is related to the Dipper-James-Mathas or to the Graham-Lehrer cellular structure).

Bip_d0,e(n) ={FLOTW bipartitions} (Jacon). Here, d₀ ≡d mod e and d₀ ∈{0,1,2, . . . ,e−1}.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 10 / 25

Kazhdan-Lusztig’s theory, Geck’s Theorem

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

R =Z[Z²]

= ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 12 / 25

Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ.

Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 12 / 25

Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰

(roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ.

Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 12 / 25

Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n→H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e.

e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ.

Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n→H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e.

e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 12 / 25

Kazhdan-Lusztig’s theory, Geck’s Theorem Cellular structures

Theorem (Kazhdan-Lusztig, 1979). For each w ∈W_n, there exists a uniqueC_w^θ ∈ H_n such that

C^θ_w =C_w^θ

C_w^θ ≡T_w mod R_<θ0

Theorem (Geck, 2007). If Lusztig’s conjectures (P1), (P2),. . . , (P14), (P15⁻) hold, then (±C_w^θ)w∈W_n

is a cellular basis of H_n.

Kazhdan-Lusztig’s theory, Geck’s Theorem Cellular structures

Theorem (Kazhdan-Lusztig, 1979). For each w ∈W_n, there exists a uniqueC_w^θ ∈ H_n such that

C^θ_w =C_w^θ

C_w^θ ≡T_w mod R_<θ0

Theorem (Geck, 2007). If Lusztig’s conjectures (P1), (P2),. . . , (P14), (P15⁻) hold, then (±C_w^θ)w∈W_n

is a cellular basis of H_n.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 13 / 25

Kazhdan-Lusztig’s theory, Geck’s Theorem Cellular structures

Case n=2 (write s =s₁)

C₁^θ = 1 C_t^θ = T_t+Q⁻¹ C_s^θ = Ts+q⁻¹

C_st^θ = T_st+Q⁻¹T_s+q⁻¹T_t+1 C_ts^θ = T_ts+Q⁻¹T_s+q⁻¹T_t+1 C_sts^θ = Tsts+q⁻¹(Tst+Tts) +

Q⁻¹q⁻¹(1+q²)T_s+q⁻²T_t+Q⁻¹q⁻² ifθ >1 Q⁻¹q⁻¹T_s+Q⁻¹q⁻¹(1−Q²)T_t+Q⁻²q⁻¹(1−Q²) if 0< θ <1 C_tst^θ = T_tst+Q⁻¹(T_st+T_ts) +

Q⁻¹q⁻¹T_s+Q⁻¹q⁻¹(1−q²)T_t+Q⁻²q⁻¹(1−q²) ifθ >1 Q⁻¹q⁻¹(1+q²)Ts+Q⁻¹q⁻¹Tt+Q⁻²q⁻¹ if 0< θ <1 C_w^θ

0 = T_w0+Q⁻¹T_sts+q⁻¹T_tst+Q⁻¹q⁻¹(T_st+T_ts) +Q⁻²q⁻¹Ts+Q⁻¹q⁻²Tt+Q⁻²q⁻²

Q⁻¹q⁻¹(1−q²) =e^(−1,−1)−e^(−1,1) ∈R_<θ0 if θ > 1

Kazhdan-Lusztig’s theory, Geck’s Theorem Cellular structures

Case n=2 (write s =s₁)

C₁^θ = 1 C_t^θ = T_t+Q⁻¹ C_s^θ = Ts+q⁻¹

C_st^θ = T_st+Q⁻¹T_s+q⁻¹T_t+1 C_ts^θ = T_ts+Q⁻¹T_s+q⁻¹T_t+1 C_sts^θ = Tsts+q⁻¹(Tst+Tts) +

Q⁻¹q⁻¹(1+q²)T_s+q⁻²T_t+Q⁻¹q⁻² ifθ >1 Q⁻¹q⁻¹T_s+Q⁻¹q⁻¹(1−Q²)T_t+Q⁻²q⁻¹(1−Q²) if 0< θ <1 C_tst^θ = T_tst+Q⁻¹(T_st+T_ts) +

Q⁻¹q⁻¹T_s+Q⁻¹q⁻¹(1−q²)T_t+Q⁻²q⁻¹(1−q²) ifθ >1 Q⁻¹q⁻¹(1+q²)Ts+Q⁻¹q⁻¹Tt+Q⁻²q⁻¹ if 0< θ <1 C_w^θ

0 = T_w0+Q⁻¹T_sts+q⁻¹T_tst+Q⁻¹q⁻¹(T_st+T_ts) +Q⁻²q⁻¹Ts+Q⁻¹q⁻²Tt+Q⁻²q⁻²

Q⁻¹q⁻¹(1−q²) =e^(−1,−1)−e^(−1,1)∈R_<θ0 if θ > 1

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 14 / 25

Conjectures

Contents

1 The set-up

Weyl group, Hecke algebra

Simple modules, decomposition map

2 Ariki’s Theorem Fock space Ariki’s Theorem

3 Kazhdan-Lusztig’s theory, Geck’s Theorem Kazhdan-Lusztig basis

Cellular structures

4 Conjectures

2-quotient, 2-core, domino tableaux Domino insertion algorithm

Conjectures, evidences

5 Comments

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

2-core of α= δ₃ = (3,2,1)

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 16 / 25

Conjectures 2-quotient, 2-core, domino tableaux

α= (5,4,4,4,4,2,1)

2-core of α= δ₃ = (3,2,1) 2-weight of α = 9

Conjectures 2-quotient, 2-core, domino tableaux

δ_r = (r,r −1, . . . ,2,1)

P_r(n) ={partitions with 2-core δ_r and 2-weight n} If α∈ P_r(n), then|α|= 1

2r(r+1) +2n

There is a map{partitions}−→{bipartitions} (called 2-quotient) inducing a bijection

π_r :P_r(n)−^∼→Bip(n)

Define the orderEr on Bip(n) by

λE^r µ⇐⇒^def π⁻¹_r (λ)Eπ⁻¹_r (µ)

If r >n−1, thenEr is the usual dominance order on bipartitions.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 17 / 25

Conjectures 2-quotient, 2-core, domino tableaux

δ_r = (r,r −1, . . . ,2,1)

P_r(n) ={partitions with 2-core δ_r and 2-weight n}

If α∈ P_r(n), then|α|= 1

2r(r+1) +2n

There is a map{partitions}−→{bipartitions} (called 2-quotient) inducing a bijection

π_r :P_r(n)−^∼→Bip(n)

Define the orderEr on Bip(n) by

λE^r µ⇐⇒^def π⁻¹_r (λ)Eπ⁻¹_r (µ)

If r >n−1, thenEr is the usual dominance order on bipartitions.

Conjectures 2-quotient, 2-core, domino tableaux

δ_r = (r,r −1, . . . ,2,1)

P_r(n) ={partitions with 2-core δ_r and 2-weight n} If α∈ P_r(n), then|α|= 1

2r(r+1) +2n

There is a map{partitions}−→{bipartitions} (called 2-quotient) inducing a bijection

π_r :P_r(n)−^∼→Bip(n)

Define the orderEr on Bip(n) by

λE^r µ⇐⇒^def π⁻¹_r (λ)Eπ⁻¹_r (µ)

If r >n−1, thenEr is the usual dominance order on bipartitions.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 17 / 25

Conjectures 2-quotient, 2-core, domino tableaux

δ_r = (r,r −1, . . . ,2,1)

P_r(n) ={partitions with 2-core δ_r and 2-weight n} If α∈ P_r(n), then|α|= 1

2r(r+1) +2n

There is a map{partitions}−→{bipartitions} (called 2-quotient)

inducing a bijection

π_r :P_r(n)−^∼→Bip(n)

Define the orderEr on Bip(n) by

λE^r µ⇐⇒^def π⁻¹_r (λ)Eπ⁻¹_r (µ)

If r >n−1, thenEr is the usual dominance order on bipartitions.

Conjectures 2-quotient, 2-core, domino tableaux

δ_r = (r,r −1, . . . ,2,1)

P_r(n) ={partitions with 2-core δ_r and 2-weight n} If α∈ P_r(n), then|α|= 1

2r(r+1) +2n

There is a map{partitions}−→{bipartitions} (called 2-quotient) inducing a bijection

π_r :P_r(n)−^∼→Bip(n)

Define the orderEr on Bip(n) by

λE^r µ⇐⇒^def π⁻¹_r (λ)Eπ⁻¹_r (µ)

If r >n−1, thenEr is the usual dominance order on bipartitions.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 17 / 25

Conjectures 2-quotient, 2-core, domino tableaux

δ_r = (r,r −1, . . . ,2,1)

P_r(n) ={partitions with 2-core δ_r and 2-weight n} If α∈ P_r(n), then|α|= 1

2r(r+1) +2n

There is a map{partitions}−→{bipartitions} (called 2-quotient) inducing a bijection

π_r :P_r(n)−^∼→Bip(n)

Define the orderEr on Bip(n) by

λEr µ⇐⇒^def π⁻¹_r (λ)Eπ⁻¹_r (µ)

If r >n−1, thenEr is the usual dominance order on bipartitions.

Conjectures 2-quotient, 2-core, domino tableaux

δ_r = (r,r −1, . . . ,2,1)

P_r(n) ={partitions with 2-core δ_r and 2-weight n} If α∈ P_r(n), then|α|= 1

2r(r+1) +2n

There is a map{partitions}−→{bipartitions} (called 2-quotient) inducing a bijection

π_r :P_r(n)−^∼→Bip(n)

Define the orderEr on Bip(n) by

λEr µ⇐⇒^def π⁻¹_r (λ)Eπ⁻¹_r (µ)

If r >n−1, thenEr is the usual dominance order on bipartitions.

C´edric Bonnaf´e (CNRS, Besan¸con, France) Cellular structures Sydney, June 2007 17 / 25