On Kazhdan-Lusztig cells in type B

On Kazhdan-Lusztig cells in type B

C´edric Bonnaf´e

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

Solstice 2008 (Paris) - June 2008

C´edric Bonnaf´e (CNRS, Besan¸con, France) Kazhdan-Lusztig cells in type B

Solstice 2008 (Paris) - June 2008 1 / 14

Weyl group

Notation

(Wn,Sn) Weyl group of type Bn

S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s1 s2 sn−1

`:Wn→N={0,1,2, . . .}length function Γ: totally ordered abelian group

ϕ:S_n−→Γ satisfyingϕ(s) =ϕ(s⁰) if s and s⁰ are conjugate in Wn, i.e.

b :=ϕ(t), a :=ϕ(s₁) =· · ·=ϕ(s_n−1). For simplification, a,b >0.

Weyl group

Notation

(Wn,Sn) Weyl group of type Bn

S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s1 s2 sn−1

`:Wn→N={0,1,2, . . .}length function Γ: totally ordered abelian group

ϕ:S_n−→Γ satisfyingϕ(s) =ϕ(s⁰) if s and s⁰ are conjugate in Wn, i.e.

b :=ϕ(t), a :=ϕ(s₁) =· · ·=ϕ(s_n−1). For simplification, a,b >0.

Weyl group

Notation

(Wn,Sn) Weyl group of type Bn

S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s1 s2 sn−1

`:Wn→N={0,1,2, . . .}length function

Γ: totally ordered abelian group

ϕ:S_n−→Γ satisfyingϕ(s) =ϕ(s⁰) if s and s⁰ are conjugate in Wn, i.e.

b :=ϕ(t), a :=ϕ(s₁) =· · ·=ϕ(s_n−1). For simplification, a,b >0.

Weyl group

Notation

(Wn,Sn) Weyl group of type Bn

S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s1 s2 sn−1

`:Wn→N={0,1,2, . . .}length function Γ: totally ordered abelian group

ϕ:S_n−→Γ satisfyingϕ(s) =ϕ(s⁰) if s and s⁰ are conjugate in Wn, i.e.

b :=ϕ(t), a :=ϕ(s₁) =· · ·=ϕ(s_n−1). For simplification, a,b >0.

Weyl group

Notation

(Wn,Sn) Weyl group of type Bn

S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s1 s2 sn−1

`:Wn→N={0,1,2, . . .}length function Γ: totally ordered abelian group

ϕ:S_n−→Γ satisfyingϕ(s) =ϕ(s⁰) if s and s⁰ are conjugate in Wn

, i.e.

b :=ϕ(t), a :=ϕ(s₁) =· · ·=ϕ(s_n−1). For simplification, a,b >0.

Weyl group

Notation

(Wn,Sn) Weyl group of type Bn

S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s1 s2 sn−1

`:Wn→N={0,1,2, . . .}length function Γ: totally ordered abelian group

ϕ:S_n−→Γ satisfyingϕ(s) =ϕ(s⁰) if s and s⁰ are conjugate in Wn, i.e.

b :=ϕ(t), a :=ϕ(s₁) =· · ·=ϕ(s_n−1).

For simplification, a,b >0.

Weyl group

Notation

(Wn,Sn) Weyl group of type Bn

S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s1 s2 sn−1

b a a a

`:Wn→N={0,1,2, . . .}length function Γ: totally ordered abelian group

ϕ:S_n−→Γ satisfyingϕ(s) =ϕ(s⁰) if s and s⁰ are conjugate in Wn, i.e.

b :=ϕ(t), a :=ϕ(s₁) =· · ·=ϕ(s_n−1).

For simplification, a,b >0.

Weyl group

Notation

(Wn,Sn) Weyl group of type Bn

S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s1 s2 sn−1

b a a a

`:Wn→N={0,1,2, . . .}length function Γ: totally ordered abelian group

ϕ:S_n−→Γ satisfyingϕ(s) =ϕ(s⁰) if s and s⁰ are conjugate in Wn, i.e.

b :=ϕ(t), a :=ϕ(s₁) =· · ·=ϕ(s_n−1).

For simplification, a,b >0.

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ]

= ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ

, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ Q =e^b, q =e^a

(Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Hecke algebra

Notation

A=Z[Γ] = ⊕

γ∈ΓZe^γ, A_<0 =Z[Γ_<0] = ⊕

γ<0Ze^γ

Q =e^b, q =e^a (Q and q are not necessarily algebraically independent)

Hecke algebra: H_n= ⊕

w∈Wn

AT_w







T_xT_y =T_xy if `(xy) =`(x) +`(y)

(T_t−Q)(T_t+Q⁻¹) =0

(T_si −q)(T_si +q⁻¹) =0 if 16i 6n−1.

H^<0_n = ⊕

w∈Wn

A_<0T_w

Involution: e^γ =e^−γ, T_w =T_w⁻¹−1

Theorem (Kazhdan-Lusztig 1979, Lusztig 1983)

If w ∈W_n, there exists a unique C_w ∈ H_n such that

C_w =C_w

C_w ≡T_w mod H^<0_n

Example. C₁ =T₁, C_t =T_t+Q⁻¹, C_si =T_si +q⁻¹ C_s1ts1 =T_s1ts1+q⁻¹(T_s1t+T_ts1) +q⁻²T_t

+Q⁻¹q⁻¹(T_s1 +q⁻¹)×







(1+q²) if b>a,

1 if b=a,

(1−Q²) if b<a

Theorem (Kazhdan-Lusztig 1979, Lusztig 1983)

If w ∈W_n, there exists a unique C_w ∈ H_n such that

C_w =C_w

C_w ≡T_w mod H^<0_n

Example. C₁ =T₁,

C_t =T_t+Q⁻¹, C_si =T_si +q⁻¹

C_s1ts1 =T_s1ts1+q⁻¹(T_s1t+T_ts1) +q⁻²T_t

+Q⁻¹q⁻¹(T_s1 +q⁻¹)×







(1+q²) if b>a,

1 if b=a,

(1−Q²) if b<a

Theorem (Kazhdan-Lusztig 1979, Lusztig 1983)

If w ∈W_n, there exists a unique C_w ∈ H_n such that

C_w =C_w

C_w ≡T_w mod H^<0_n

Example. C₁ =T₁, C_t =T_t+Q⁻¹, C_si =T_si +q⁻¹

C_s1ts1 =T_s1ts1+q⁻¹(T_s1t+T_ts1) +q⁻²T_t

+Q⁻¹q⁻¹(T_s1 +q⁻¹)×







(1+q²) if b>a,

1 if b=a,

(1−Q²) if b<a

Theorem (Kazhdan-Lusztig 1979, Lusztig 1983)

If w ∈W_n, there exists a unique C_w ∈ H_n such that

C_w =C_w

C_w ≡T_w mod H^<0_n

Example. C₁ =T₁, C_t =T_t+Q⁻¹, C_si =T_si +q⁻¹ C_s1ts1 =T_s1ts1+q⁻¹(T_s1t+T_ts1) +q⁻²T_t

+Q⁻¹q⁻¹(T_s1 +q⁻¹)×







(1+q²) if b>a,

1 if b=a,

(1−Q²) if b<a

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0)

,A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹]

One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ

: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n:

works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space)

Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory

Jacon-B.

Two different situations:

(1) Γ =Z, v =e¹, A=Z[v,v⁻¹], a,b ∈Z>0.

(2) Γ =Z², a= (0,1), b = (1,0),A=Z[Q,Q⁻¹,q,q⁻¹] One can then choose different orders onΓ: for instance, if

ξ∈R>0\Q, one can define 6ξ on Z² by

(m,n)6ξ (0,0)⇐⇒mξ+n60.

Remarks. • Situation (1) occurs in representation theory of finite reductive groups with 2b/a ∈Z

•General situation (1) AND situation (2) arise in the study of modular representations of H_n: works of

Ariki (decomposition matrices and Fock space) Uglov, Jacon, Geck-Jacon: parametrization of simple kHn-modules, canonical sets

Geck: cellular structures onH_n coming from KL-theory Jacon-B.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LCAC_x I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−:

it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LCAC_x I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LCAC_x I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LCAC_x I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LCAC_x I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LC

AC_x

I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LC

AC_x I_<LC = ⊕

x<LCAC_x

VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LC

AC_x I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LC

AC_x I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module

: VC is called the left cell representation associated toC.

If x, y ∈W_n, we write x ←^L−y if there exists h∈ H_n such that Cx occurs in hCy

Let6L be the transitive closure of←^L−: it is a preorder (reflexive and transitive)

Let∼L be the equivalence relation associated to6^L (i.e. x ∼Ly if and only ifx 6L y and y 6Lx)

Definition

Aleft cell is an equivalence class for the relation ∼L.

If C is a left cell, we set









I_6LC = ⊕

x6LC

AC_x I_<LC = ⊕

x<LCAC_x VC =I_6LC/I_<LC

By construction, I_6LC and I_<LC are left ideals of H_n and VC is a leftH_n-module: VC is called the left cell representation associated toC.

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹, so x 6L y ⇐⇒x⁻¹6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases (for instance in the symmetric group using the Robinson-Schensted correspondence: Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group). The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹, so x 6L y ⇐⇒x⁻¹6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases (for instance in the symmetric group using the Robinson-Schensted correspondence: Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group). The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹

, so x 6L y ⇐⇒x⁻¹6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases (for instance in the symmetric group using the Robinson-Schensted correspondence: Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group). The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹, so x 6L y ⇐⇒x⁻¹ 6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases (for instance in the symmetric group using the Robinson-Schensted correspondence: Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group). The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹, so x 6L y ⇐⇒x⁻¹ 6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases (for instance in the symmetric group using the Robinson-Schensted correspondence: Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group). The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹, so x 6L y ⇐⇒x⁻¹ 6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases

(for instance in the symmetric group using the Robinson-Schensted correspondence: Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group). The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹, so x 6L y ⇐⇒x⁻¹ 6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases (for instance in the symmetric group using the Robinson-Schensted correspondence:

Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group). The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹, so x 6L y ⇐⇒x⁻¹ 6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases (for instance in the symmetric group using the Robinson-Schensted correspondence:

Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group).

The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Comments

One could definex ←^R−y (“C_x occurs in some C_yh”) or x ←^LR−y (“C_x occurs in some hC_yh⁰”)

This leads to6^R, 6^LR,∼R and ∼LR, right/two-sidedcells.

I The anti-automorphismT_x 7→T_x⁻¹ sendsC_x to C_x⁻¹, so x 6L y ⇐⇒x⁻¹ 6R y⁻¹

I Lusztigconjectures that∼LR is generated by∼L and∼R.

Remark

The relations∼? can be computed in some cases (for instance in the symmetric group using the Robinson-Schensted correspondence:

Kazhdan-Lusztig 1979).

However, the preorder6L or 6R is in general unknown (even in the symmetric group). The preorder 6LR seems to be easier (for

instance, it is given by the dominance order on partitions through the Robinson-Schensted correspondence).

Domino insertion algorithm:

We identify W_n with the group of permutationsw of I_n ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and s_i →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

• • •

• •

•

Domino insertion algorithm:

We identify W_n with the group of permutationsw of I_n ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and si →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W9

↑

• • •

• •

•

Domino insertion algorithm:

We identify W_n with the group of permutationsw of I_n ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and si →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W9

↑

• • • 7

• •

•

Domino insertion algorithm:

We identify W_n with the group of permutationsw of I_n ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and si →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W9

↑

• • • 7

• •

•

Domino insertion algorithm:

We identify W_n with the group of permutationsw of In ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and s_i →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 7

• •

• 8

Domino insertion algorithm:

We identify W_n with the group of permutationsw of In ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and s_i →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 7

• •

• 8

Domino insertion algorithm:

We identify W_n with the group of permutationsw of I_n ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and s_i →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 7

• •

• 8 9

Domino insertion algorithm:

We identify W_n with the group of permutationsw of I_n ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and s_i →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 7

• •

• 8 9

Domino insertion algorithm:

We identify W_n with the group of permutationsw of I_n ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and s_i →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1

• • 7

• 8 9

Domino insertion algorithm:

We identify W_n with the group of permutationsw of I_n ={±1,±2, . . . ,±n}such that w(−i) = −w(i) through

t 7→(−1,1) and s_i →7 (i,i +1)(−i,−i −1) Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1

• • 7

• 8 9