Semicontinuity properties of Kazhdan-Lusztig cells

Semicontinuity properties of Kazhdan-Lusztig cells

C´edric Bonnaf´e

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

MSRI (Berkeley) - March 2008

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we write s ∼s⁰ if s and s⁰ are conjugate in W ϕ:S →Γ, where Γ is an ordered group and ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials/representations are involved in:

Representations of reductive groups and Lie algebras Geometry of flag varieties (intersection cohomology) ...

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we writes ∼s⁰ if s and s⁰ are conjugate in W

ϕ:S →Γ, where Γ is an ordered group and ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials/representations are involved in:

Representations of reductive groups and Lie algebras Geometry of flag varieties (intersection cohomology) ...

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we writes ∼s⁰ if s and s⁰ are conjugate in W ϕ:S →Γ, where Γ is an ordered group

and ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials/representations are involved in:

Representations of reductive groups and Lie algebras Geometry of flag varieties (intersection cohomology) ...

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we writes ∼s⁰ if s and s⁰ are conjugate in W ϕ:S →Γ, where Γ is an ordered group and ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials/representations are involved in:

Representations of reductive groups and Lie algebras Geometry of flag varieties (intersection cohomology) ...

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we writes ∼s⁰ if s and s⁰ are conjugate in W ϕ:S →Γ, where Γ is an ordered group and ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials/representations are involved in:

Representations of reductive groups and Lie algebras Geometry of flag varieties (intersection cohomology) ...

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we writes ∼s⁰ if s and s⁰ are conjugate in W ϕ:S →Γ, where Γ is an ordered group and ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials/representations are involved in:

Representations of reductive groups and Lie algebras

Geometry of flag varieties (intersection cohomology) ...

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we writes ∼s⁰ if s and s⁰ are conjugate in W ϕ:S →Γ, where Γ is an ordered group and ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials/representations are involved in:

Representations of reductive groups and Lie algebras Geometry of flag varieties (intersection cohomology)

...

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we writes ∼s⁰ if s and s⁰ are conjugate in W ϕ:S →Γ, where Γ is an ordered group and ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials/representations are involved in:

Representations of reductive groups and Lie algebras Geometry of flag varieties (intersection cohomology) ...

Notation

(W,S) Coxeter group

If s, s⁰ ∈S, we writes ∼s⁰ if s and s⁰ are conjugate in W ϕ:S →Rand ϕ(s) =ϕ(s⁰) if s ∼s⁰.

To this datum, Kazhdan and Lusztig have associated a partition of W into (left, right, two-sided)cells, and representations of the associated Hecke algebra.

Remark. If W is a finite or anaffine Weyl group, then KL-cells/polynomials are involved in:

Representations of reductive groups and Lie algebras Geometry of flag varieties (intersection cohomology) ...

Contents

Kazhdan-Lusztig cells

Conjectures for typeB (joint with Geck-Iancu-Lam) Conjectures for general Coxeter groups (finite or not)

• F4 (Geck)

• Affine Weyl groups (Guilhot)

Contents

Kazhdan-Lusztig cells

Conjectures for typeB (joint with Geck-Iancu-Lam)

Conjectures for general Coxeter groups (finite or not)

• F4 (Geck)

• Affine Weyl groups (Guilhot)

Contents

Kazhdan-Lusztig cells

Conjectures for typeB (joint with Geck-Iancu-Lam) Conjectures for general Coxeter groups (finite or not)

• F4 (Geck)

• Affine Weyl groups (Guilhot)

Contents

Kazhdan-Lusztig cells

Conjectures for typeB (joint with Geck-Iancu-Lam) Conjectures for general Coxeter groups (finite or not)

• F4 (Geck)

• Affine Weyl groups (Guilhot)

Contents

Kazhdan-Lusztig cells

Conjectures for typeB (joint with Geck-Iancu-Lam) Conjectures for general Coxeter groups (finite or not)

• F4 (Geck)

• Affine Weyl groups (Guilhot)

Kazhdan-Lusztig cells

Notation

A=Z[R] = ⊕

γ∈RZv^γ,K =Frac(A) A_<0 =Z[R<0] = ⊕

γ<0Zv^γ Hecke algebra: H= ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s −v^ϕ(s))(T_s +v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Kazhdan-Lusztig cells

Notation

A=Z[R]

= ⊕

γ∈RZv^γ,K =Frac(A) A_<0 =Z[R<0] = ⊕

γ<0Zv^γ Hecke algebra: H= ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s −v^ϕ(s))(T_s +v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Kazhdan-Lusztig cells

Notation

A=Z[R] = ⊕

γ∈RZv^γ

,K =Frac(A) A_<0 =Z[R<0] = ⊕

γ<0Zv^γ Hecke algebra: H= ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s −v^ϕ(s))(T_s +v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Kazhdan-Lusztig cells

Notation

A=Z[R] = ⊕

γ∈RZv^γ,K =Frac(A)

A_<0 =Z[R<0] = ⊕

γ<0Zv^γ Hecke algebra: H= ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s −v^ϕ(s))(T_s +v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Kazhdan-Lusztig cells

Notation

A=Z[R] = ⊕

γ∈RZv^γ,K =Frac(A) A_<0 =Z[R<0] = ⊕

γ<0Zv^γ

Hecke algebra: H= ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s −v^ϕ(s))(T_s +v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Kazhdan-Lusztig cells

Notation

A=Z[R] = ⊕

γ∈RZv^γ,K =Frac(A) A_<0 =Z[R<0] = ⊕

γ<0Zv^γ Hecke algebra: H = ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s −v^ϕ(s))(T_s +v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Kazhdan-Lusztig cells

Notation

A=Z[R] = ⊕

γ∈RZv^γ,K =Frac(A) A_<0 =Z[R<0] = ⊕

γ<0Zv^γ Hecke algebra: H = ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s−v^ϕ(s))(T_s+v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function

H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Kazhdan-Lusztig cells

Notation

A=Z[R] = ⊕

γ∈RZv^γ,K =Frac(A) A_<0 =Z[R<0] = ⊕

γ<0Zv^γ Hecke algebra: H = ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s−v^ϕ(s))(T_s+v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Kazhdan-Lusztig cells

Notation

A=Z[R] = ⊕

γ∈RZv^γ,K =Frac(A) A_<0 =Z[R<0] = ⊕

γ<0Zv^γ Hecke algebra: H = ⊕

w∈WAT_w

T_xT_y =T_xy if `(xy) =`(x) +`(y) (T_s−v^ϕ(s))(T_s+v^−ϕ(s)) =0 if s ∈S

where`:W →N={0,1,2,3, . . .} is the length function H<0 = ⊕

w∈WA<0Tw

Involution: v^γ =v^−γ, T_w =T_w⁻¹−1

Theorem (Kazhdan-Lusztig 1979, Lusztig 1983)

If w ∈W , there exists a unique C_w ∈ H such that

C_w =C_w

C_w ≡T_w mod H_<0

Example. C₁ =T₁

C_s =







T_s+v^−ϕ(s) if ϕ(s)>0

T_s if ϕ(s) =0

T_s−v^ϕ(s) if ϕ(s)<0

Theorem (Kazhdan-Lusztig 1979, Lusztig 1983)

If w ∈W , there exists a unique C_w ∈ H such that

C_w =C_w

C_w ≡T_w mod H_<0

Example. C₁ =T₁

C_s =







T_s+v^−ϕ(s) if ϕ(s)>0

T_s if ϕ(s) =0

T_s−v^ϕ(s) if ϕ(s)<0

Theorem (Kazhdan-Lusztig 1979, Lusztig 1983)

If w ∈W , there exists a unique C_w ∈ H such that

C_w =C_w

C_w ≡T_w mod H_<0

Example. C₁ =T₁

C_s =







Ts+v^−ϕ(s) if ϕ(s)>0

Ts if ϕ(s) =0

Ts−v^ϕ(s) if ϕ(s)<0

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-module

: VC is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L−y if there existsh ∈ H 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 and VC is a leftH-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).

An “easy” remark

LetSϕ={s ∈S | ϕ(s) =0}and Wϕ =hSϕi.

LetI =S\S_ϕ,˜I ={wtw⁻¹ |w ∈W_ϕ, t ∈I},and W˜ =h˜Ii.

Theorem (Dyer)

(W˜,˜I) is a Coxeter group and W =W_ϕnW .˜

Corollary

H(W,S, ϕ) =W_ϕnH(W˜,˜I,ϕ), where˜ ϕ(wtw˜ ⁻¹) =ϕ(t) (w ∈W_ϕ, t ∈I ).

Corollary

Since C_s =T_s and C_sw =C_sC_w for all s ∈S_ϕ and w ∈W , the left cells of(W,S, ϕ) are of the form W_ϕ· C, where C is a left cell of (W˜,˜I,ϕ).˜

An “easy” remark

LetSϕ={s ∈S | ϕ(s) =0}and Wϕ =hSϕi.

LetI =S\S_ϕ,˜I ={wtw⁻¹ |w ∈W_ϕ, t ∈I},and W˜ =h˜Ii.

Theorem (Dyer)

(W˜,˜I) is a Coxeter group and W =W_ϕnW .˜

Corollary

H(W,S, ϕ) =W_ϕnH(W˜,˜I,ϕ), where˜ ϕ(wtw˜ ⁻¹) =ϕ(t) (w ∈W_ϕ, t ∈I ).

Corollary

Since C_s =T_s and C_sw =C_sC_w for all s ∈S_ϕ and w ∈W , the left cells of(W,S, ϕ) are of the form W_ϕ· C, where C is a left cell of (W˜,˜I,ϕ).˜

An “easy” remark

LetSϕ={s ∈S | ϕ(s) =0}and Wϕ =hSϕi.

LetI =S\S_ϕ

,˜I ={wtw⁻¹ |w ∈W_ϕ, t ∈I},and W˜ =h˜Ii.

Theorem (Dyer)

(W˜,˜I) is a Coxeter group and W =W_ϕnW .˜

Corollary

H(W,S, ϕ) =W_ϕnH(W˜,˜I,ϕ), where˜ ϕ(wtw˜ ⁻¹) =ϕ(t) (w ∈W_ϕ, t ∈I ).

Corollary

Since C_s =T_s and C_sw =C_sC_w for all s ∈S_ϕ and w ∈W , the left cells of(W,S, ϕ) are of the form W_ϕ· C, where C is a left cell of (W˜,˜I,ϕ).˜

An “easy” remark

LetSϕ={s ∈S | ϕ(s) =0}and Wϕ =hSϕi.

LetI =S\S_ϕ,˜I ={wtw⁻¹ |w ∈W_ϕ, t ∈I},and W˜ =h˜Ii.

Theorem (Dyer)

(W˜,˜I) is a Coxeter group and W =W_ϕnW .˜

Corollary

H(W,S, ϕ) =W_ϕnH(W˜,˜I,ϕ), where˜ ϕ(wtw˜ ⁻¹) =ϕ(t) (w ∈W_ϕ, t ∈I ).

Corollary

Since C_s =T_s and C_sw =C_sC_w for all s ∈S_ϕ and w ∈W , the left cells of(W,S, ϕ) are of the form W_ϕ· C, where C is a left cell of (W˜,˜I,ϕ).˜

An “easy” remark

LetSϕ={s ∈S | ϕ(s) =0}and Wϕ =hSϕi.

LetI =S\S_ϕ,˜I ={wtw⁻¹ |w ∈W_ϕ, t ∈I},and W˜ =h˜Ii.

Theorem (Dyer)

(W˜,˜I) is a Coxeter group and W =W_ϕnW .˜

Corollary

H(W,S, ϕ) =W_ϕnH(W˜,˜I,ϕ), where˜ ϕ(wtw˜ ⁻¹) =ϕ(t) (w ∈W_ϕ, t ∈I ).

Corollary

Since C_s =T_s and C_sw =C_sC_w for all s ∈S_ϕ and w ∈W , the left cells of(W,S, ϕ) are of the form W_ϕ· C, where C is a left cell of (W˜,˜I,ϕ).˜

An “easy” remark

LetSϕ={s ∈S | ϕ(s) =0}and Wϕ =hSϕi.

LetI =S\S_ϕ,˜I ={wtw⁻¹ |w ∈W_ϕ, t ∈I},and W˜ =h˜Ii.

Theorem (Dyer)

(W˜,˜I) is a Coxeter group and W =W_ϕnW .˜

Corollary

H(W,S, ϕ) =W_ϕnH(W˜,˜I,ϕ), where˜ ϕ(wtw˜ ⁻¹) =ϕ(t) (w ∈W_ϕ, t ∈I ).

Corollary

Since C_s =T_s and C_sw =C_sC_w for all s ∈S_ϕ and w ∈W , the left cells of(W,S, ϕ) are of the form W_ϕ· C, where C is a left cell of (W˜,˜I,ϕ).˜

An “easy” remark

LetSϕ={s ∈S | ϕ(s) =0}and Wϕ =hSϕi.

LetI =S\S_ϕ,˜I ={wtw⁻¹ |w ∈W_ϕ, t ∈I},and W˜ =h˜Ii.

Theorem (Dyer)

(W˜,˜I) is a Coxeter group and W =W_ϕnW .˜

Corollary

H(W,S, ϕ) =W_ϕnH(W˜,˜I,ϕ), where˜ ϕ(wtw˜ ⁻¹) =ϕ(t) (w ∈W_ϕ, t ∈I ).

Corollary

Since C_s =T_s and C_sw =C_sC_w for all s ∈S_ϕ and w ∈W , the left cells of(W,S, ϕ) are of the form Wϕ· C, where C is a left cell of (W˜,˜I,ϕ).˜

Examples

TypeB

i i i · · · ⁱ

t s₁ s₂ s_n−1

W(B_n) = S_n×(Z/2Z)ⁿ =htinW(D_n) TypeF₄

i i i i

t2 t1 s1 s2

W(F₄)

=

=

i i

t2 t1

S₃ n n

i i i i

""

bb

s1 s2 t₁s₁t₁ t₂t₁s₁t₁t₂ W(D₄)

Examples

TypeB

i i i · · · ⁱ

t s₁ s₂ s_n−1

W(B_n) = S_n×(Z/2Z)ⁿ

=htinW(D_n) TypeF₄

i i i i

t2 t1 s1 s2

W(F₄)

=

=

i i

t2 t1

S₃ n n

i i i i

""

bb

s1 s2 t₁s₁t₁ t₂t₁s₁t₁t₂ W(D₄)

Examples

TypeB

i i i · · · ⁱ

t s₁ s₂ s_n−1

W(B_n) = S_n×(Z/2Z)ⁿ =htinW(D_n)

TypeF₄

i i i i

t2 t1 s1 s2

W(F₄)

=

=

i i

t2 t1

S₃ n n

i i i i

""

bb

s1 s2 t₁s₁t₁ t₂t₁s₁t₁t₂ W(D₄)

Examples

TypeB

i i i · · · ⁱ

t s₁ s₂ s_n−1

W(B_n) = S_n×(Z/2Z)ⁿ =htinW(D_n) TypeF₄

i i i i

t2 t1 s1 s2

W(F₄)

=

=

i i

t2 t1

S₃ n n

i i i i

""

bb

s1 s2 t₁s₁t₁ t₂t₁s₁t₁t₂ W(D₄)

Examples

TypeB

i i i · · · ⁱ

t s₁ s₂ s_n−1

W(B_n) = S_n×(Z/2Z)ⁿ =htinW(D_n) TypeF₄

i i i i

t2 t1 s1 s2

W(F₄)

=

=

i i

t2 t1

S₃ n n

i i i i

""

bb

s1 s2 t₁s₁t₁ t₂t₁s₁t₁t₂ W(D₄)

Examples

TypeB

i i i · · · ⁱ

t s₁ s₂ s_n−1

W(B_n) = S_n×(Z/2Z)ⁿ =htinW(D_n) TypeF₄

i i i i

t2 t1 s1 s2

W(F₄)

=

=

i i

t2 t1

S₃ n n

i i i i

""

bb

s1 s2 t₁s₁t₁ t₂t₁s₁t₁t₂ W(D₄)

Type B

(W,S) = (W_n,S_n), where S_n={t,s₁,s₂, . . . ,s_n−1}and

i i i · · · ⁱ

t s₁ s₂ s_n−1

ϕ−→ b a a a

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)

Type B

(W,S) = (W_n,S_n), where S_n={t,s₁,s₂, . . . ,s_n−1}and

i i i · · · ⁱ

t s₁ s₂ s_n−1

ϕ−→ b a a a

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)

Type B

(W,S) = (W_n,S_n), where S_n={t,s₁,s₂, . . . ,s_n−1}and

i i i · · · ⁱ

t s₁ s₂ s_n−1

ϕ−→ b a a a

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)

Type B

(W,S) = (W_n,S_n), where S_n={t,s₁,s₂, . . . ,s_n−1}and

i i i · · · ⁱ

t s₁ s₂ s_n−1

ϕ−→ b a a a

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)

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W9

• • •

• •

•

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • •

• •

•

Domino insertion algorithm:

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:

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:

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:

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:

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:

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:

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:

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:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 6

• • 7

• 8 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 6

• • 7

• 8 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 6

• • 7

• 8 4 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 6

• • 7

• 8 4 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 5

• • 7

• 8 4 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 5

• • 6

• 8 7 4

9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 5

• • 6

• 7

4 8

9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 5

• • 6

• 7

4 8

9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 3

• • 6

• 7

4 8

9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 3

• • 5

• 7

4 8

9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 3

• • 5

• 6

4 8

9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 3

• • 5

• 6

4 7

8 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 3

• • 5

• 6

4 7

8 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 3

• • 5

• 6

2 7

8 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 3

• • 5

• 4 6

2 7

9 8

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

↑

• • • 1 3

• • 5

• 4 6

2 7

8 9

Domino insertion algorithm:

Example: w =

1 2 3 4 5 6 7 8 9

7 −8 −9 1 6 −4 5 3 −2

∈W₉

D₃(w) =

• • • 1 3

• • 5

• 4 6

2 7

8 9