Representation theory of the Mantaci-Reutenauer algebra

Representation theory of the Mantaci-Reutenauer algebra

C´ edric Bonnaf´ e

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

Groups in Galway, May 2006

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Solomon’s descent algebra

Let (W , S ) be a finite Coxeter group:

W =< S|∀ s, s ⁰ ∈ S, s ² = (ss ⁰ ) ^m

= 1 >

Length function: ` : W → N

Parabolic subgroups: I ⊂ S, W _I =< I >, (W _I , I ) is a finite Coxeter group.

X _I = {x ∈ W | ∀ w ∈ W _I , `(xw ) > `(x )} X _I − → ^∼ W /W _I

x 7→ xW _I

Solomon’s descent algebra

Let (W , S ) be a finite Coxeter group:

W =< S|∀ s, s ⁰ ∈ S, s ² = (ss ⁰ ) ^m

= 1 >

Length function: ` : W → N

Parabolic subgroups: I ⊂ S, W _I =< I >, (W _I , I ) is a finite Coxeter group.

X _I = {x ∈ W | ∀ w ∈ W _I , `(xw ) > `(x )} X _I − → ^∼ W /W _I

x 7→ xW _I

Solomon’s descent algebra

Let (W , S ) be a finite Coxeter group:

W =< S|∀ s, s ⁰ ∈ S, s ² = (ss ⁰ ) ^m

= 1 >

Length function: ` : W → N

Parabolic subgroups: I ⊂ S, W _I =< I >, (W _I , I ) is a finite Coxeter group.

X _I = {x ∈ W | ∀ w ∈ W _I , `(xw ) > `(x )}

X _I − → ^∼ W /W _I

x 7→ xW _I

Solomon’s descent algebra

Let X IJ = (X I ) ⁻¹ ∩ X J

X _IJ − → ^∼ W _I \W /W _J d 7→ W _I dW _J

If d ∈ X _IJ , then W _I ∩ ^d W _J = W _I∩

_J X _J = a

d∈X

X _I∩ ^I

J .d

Solomon (1976): x _I = X

w∈X

w ∈ Z W

Σ(W ) := ⊕

I ⊂S Z x _I ⊂ Z W θ : Σ(W ) −→ Z Irr W

x _I 7−→ Ind ^W _W

I

1 _I x _I x _J = X

d∈X

x _I x _I ^I _∩

J d = X

d∈X

x _I∩

J d = X

d∈X

x

I∩J

Solomon’s descent algebra

Let X IJ = (X I ) ⁻¹ ∩ X J

X _IJ − → ^∼ W _I \W /W _J d 7→ W _I dW _J

If d ∈ X _IJ , then W _I ∩ ^d W _J = W _I∩

_J X _J = a

d∈X

X _I∩ ^I

J .d

Solomon (1976): x I = X

w∈X

w ∈ Z W

Σ(W ) := ⊕

I⊂S Z x I ⊂ Z W θ : Σ(W ) −→ Z Irr W

x _I 7−→ Ind ^W _W

I

1 _I

x _I x _J = X

d∈X

x _I x _I ^I _∩

J d = X

d∈X

x _I∩

J d = X

d∈X

x

I∩J

Solomon’s descent algebra

Let X IJ = (X I ) ⁻¹ ∩ X J

X _IJ − → ^∼ W _I \W /W _J d 7→ W _I dW _J

If d ∈ X _IJ , then W _I ∩ ^d W _J = W _I∩

_J X _J = a

d∈X

X _I∩ ^I

J .d

Solomon (1976): x I = X

w∈X

w ∈ Z W

Σ(W ) := ⊕

I⊂S Z x I ⊂ Z W θ : Σ(W ) −→ Z Irr W

x _I 7−→ Ind ^W _W

I

1 _I x _I x _J = X

d∈X

x _I x _I ^I _∩

J d = X

d∈X

x _I∩

J d = X

d∈X

x

I∩J

Solomon’s descent algebra

Theorem (Solomon).

Σ(W ) is a subalgebra of Z W . θ is a morphism of algebras.

Ker θ = X

I ≡J

Z (x _I − x _J ).

Q Ker θ is the radical of Q Σ(W ).

θ(w 0 ) = ε.

I ≡ J ⇔ W _I ∼ W _J .

w ₀ is the longest element of W

Solomon’s descent algebra

Further works:

Idempotents (Bergeron-Bergeron-Howlett-Taylor) Cartan matrix unitriangular (?)

Modular representations (Atkinson-Pfeiffer-Van Willigenburg):

simples, radical...

Lie idempotents (Reutenauer, Erdmann-Schocker) Symmetry property (Blessenhohl-Hohlweg-Schocker):

θ(x _I )(x _J ) = θ(x _J )(x _I )

Complex reflection groups (Mathas)

Loewy length (B.-Pfeiffer): all cases except type D _2n+1 ...

(The Loewy length of an algebra A is the minimal natural number

k > 1 such that (Rad A) ^k = 0)

Solomon’s descent algebra

Further works:

Idempotents (Bergeron-Bergeron-Howlett-Taylor) Cartan matrix unitriangular (?)

Modular representations (Atkinson-Pfeiffer-Van Willigenburg):

simples, radical...

Lie idempotents (Reutenauer, Erdmann-Schocker) Symmetry property (Blessenhohl-Hohlweg-Schocker):

θ(x _I )(x _J ) = θ(x _J )(x _I )

Complex reflection groups (Mathas)

Loewy length (B.-Pfeiffer): all cases except type D _2n+1 ...

(The Loewy length of an algebra A is the minimal natural number

k > 1 such that (Rad A) ^k = 0)

Solomon’s descent algebra

Theorem (B.-Pfeiffer, 2005). Let σ denote the automorphism

of W induced by conjugacy by w ₀ . Then the Loewy length of

Q Σ(W ) ^σ is d|S|/2e.

Solomon’s descent algebra

Problem: θ is surjective if and only if W is a product of symmetric

groups.

Mantaci-Reutenauer algebra Coxeter group of type B

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Mantaci-Reutenauer algebra Coxeter group of type B

Let (W _n , S _n ) be a Coxeter group of type B _n .

i i i · · · ⁱ

t s ₁ s ₂ s n−1

t ₁ = t , t _{i +1} = s _i t _i s _i

I _n = {1, 2, . . . , n} ∪ {−1, −2, . . . , −n} W _n = {σ : I _n − → ^∼ I _n | ∀ i ∈ I _n , σ(−i ) = −σ(i )}

s _i = (i , i + 1)(−i , −i − 1) t _i = (i , −i)

Let T _n : Z W _n → Z : canonical symmetrizing form

Mantaci-Reutenauer algebra Coxeter group of type B

Let (W _n , S _n ) be a Coxeter group of type B _n .

i i i · · · ⁱ

t s ₁ s ₂ s n−1

t ₁ = t , t _{i +1} = s _i t _i s _i

I _n = {1, 2, . . . , n} ∪ {−1, −2, . . . , −n} W _n = {σ : I _n − → ^∼ I _n | ∀ i ∈ I _n , σ(−i ) = −σ(i )}

s _i = (i , i + 1)(−i , −i − 1) t _i = (i , −i)

Let T _n : Z W _n → Z : canonical symmetrizing form

Mantaci-Reutenauer algebra Coxeter group of type B

Let (W _n , S _n ) be a Coxeter group of type B _n .

i i i · · · ⁱ

t s ₁ s ₂ s n−1

t ₁ = t , t _{i +1} = s _i t _i s _i

I _n = {1, 2, . . . , n} ∪ {−1, −2, . . . , −n} W _n = {σ : I _n − → ^∼ I _n | ∀ i ∈ I _n , σ(−i ) = −σ(i )}

s _i = (i , i + 1)(−i , −i − 1) t _i = (i , −i)

Let T _n : Z W _n → Z : canonical symmetrizing form

Mantaci-Reutenauer algebra Coxeter group of type B

Let (W _n , S _n ) be a Coxeter group of type B _n .

i i i · · · ⁱ

t s ₁ s ₂ s n−1

t ₁ = t , t _{i +1} = s _i t _i s _i

I _n = {1, 2, . . . , n} ∪ {−1, −2, . . . , −n}

W _n = {σ : I _n − → ^∼ I _n | ∀ i ∈ I _n , σ(−i ) = −σ(i )}

s _i = (i , i + 1)(−i , −i − 1) t _i = (i , −i)

Let T _n : Z W _n → Z : canonical symmetrizing form

Mantaci-Reutenauer algebra Coxeter group of type B

Let (W _n , S _n ) be a Coxeter group of type B _n .

i i i · · · ⁱ

t s ₁ s ₂ s n−1

t ₁ = t , t _{i +1} = s _i t _i s _i

I _n = {1, 2, . . . , n} ∪ {−1, −2, . . . , −n}

W _n = {σ : I _n − → ^∼ I _n | ∀ i ∈ I _n , σ(−i ) = −σ(i )}

s _i = (i , i + 1)(−i , −i − 1) t _i = (i , −i)

Let T _n : Z W _n → Z : canonical symmetrizing form

Mantaci-Reutenauer algebra Some reflection subgroups

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Mantaci-Reutenauer algebra Some reflection subgroups

S −n = {s 1 , . . . , s n−1 }, W −n =< S −n >' S n

S _n ⁰ = S _n ∪ {t ₁ , . . . , t _n }

W _n = S _n n < t ₁ , . . . , t _n >

Comp(n) = {signed compositions of n}

| Comp(n)| = 2.3 ⁿ⁻¹ C = (c 1 , . . . , c r ), |c 1 | + · · · + |c r | = n

W _C ' W _c

× · · · × W _c

⊂ W _n

S _C ⁰ = S _n ⁰ ∩ W _C ⇒ W _C =< S _C ⁰ >

Mantaci-Reutenauer algebra Some reflection subgroups

S −n = {s 1 , . . . , s n−1 }, W −n =< S −n >' S n

S _n ⁰ = S _n ∪ {t ₁ , . . . , t _n }

W _n = S _n n < t ₁ , . . . , t _n >

Comp(n) = {signed compositions of n}

| Comp(n)| = 2.3 ⁿ⁻¹ C = (c 1 , . . . , c r ), |c 1 | + · · · + |c r | = n

W _C ' W _c

× · · · × W _c

⊂ W _n

S _C ⁰ = S _n ⁰ ∩ W _C ⇒ W _C =< S _C ⁰ >

Mantaci-Reutenauer algebra Some reflection subgroups

S −n = {s 1 , . . . , s n−1 }, W −n =< S −n >' S n

S _n ⁰ = S _n ∪ {t ₁ , . . . , t _n }

W _n = S _n n < t ₁ , . . . , t _n >

Comp(n) = {signed compositions of n}

| Comp(n)| = 2.3 ⁿ⁻¹ C = (c 1 , . . . , c r ), |c 1 | + · · · + |c r | = n

W _C ' W _c

× · · · × W _c

⊂ W _n

S _C ⁰ = S _n ⁰ ∩ W _C ⇒ W _C =< S _C ⁰ >

Mantaci-Reutenauer algebra Definition

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Mantaci-Reutenauer algebra Definition

X _C = {x ∈ W _n | ∀ w ∈ W _C , `(xw ) > `(x )}

X _C − → ^∼ W _n /W _C x 7→ xW _C

X _CD = (X _C ) ⁻¹ ∩ X _D X CD

− → ∼ W C \W n /W D

d 7→ W _C dW _D

If d ∈ X _CD , then W _C ∩ ^d W _D = W _C∩

D

BUT (in general)

X _D 6= a

d∈X

X _C ^C _∩

D .d

Mantaci-Reutenauer algebra Definition

X _C = {x ∈ W _n | ∀ w ∈ W _C , `(xw ) > `(x )}

X _C − → ^∼ W _n /W _C x 7→ xW _C X _CD = (X _C ) ⁻¹ ∩ X _D

X CD

− → ∼ W C \W n /W D

d 7→ W _C dW _D

If d ∈ X CD , then W C ∩ ^d W D = W C∩

D

BUT (in general)

X _D 6= a

d∈X

X _C ^C _∩

D .d

Mantaci-Reutenauer algebra Definition

X _C = {x ∈ W _n | ∀ w ∈ W _C , `(xw ) > `(x )}

X _C − → ^∼ W _n /W _C x 7→ xW _C X _CD = (X _C ) ⁻¹ ∩ X _D

X CD

− → ∼ W C \W n /W D

d 7→ W _C dW _D

If d ∈ X CD , then W C ∩ ^d W D = W C∩

D

BUT (in general)

X _D 6= a

d∈X

X _C ^C _∩

D .d

Mantaci-Reutenauer algebra Definition

Let x C = X

w∈X

w ∈ Z W n

Let Σ ⁰ (W n ) := ⊕

C ∈Comp(n) Z x C ⊂ Z W n

Let

θ ⁰ : Σ ⁰ (W _n ) −→ Z Irr W _n x _C 7−→ Ind ^W _W

C

1 _C

Theorem (B.-Hohlweg, 2004). Σ ⁰ (W n ) is a subalgebra of Z W n . θ ⁰ is a surjective morphism of algebras. Ker θ ⁰ = X

C≡D

Z (x _C − x _D ).

Q Ker θ ⁰ is the radical of Q Σ ⁰ (W _n ).

Σ ⁰ (W _n ) ' Mantaci-Reutenauer algebra.

T n (xy ) = hθ ⁰ (x ), θ ⁰ (y )i.

Mantaci-Reutenauer algebra Definition

Let x C = X

w∈X

w ∈ Z W n

Let Σ ⁰ (W n ) := ⊕

C ∈Comp(n) Z x C ⊂ Z W n

Let

θ ⁰ : Σ ⁰ (W _n ) −→ Z Irr W _n x _C 7−→ Ind ^W _W

C

1 _C

Theorem (B.-Hohlweg, 2004).

Σ ⁰ (W n ) is a subalgebra of Z W n . θ ⁰ is a surjective morphism of algebras.

Ker θ ⁰ = X

C≡D

Z (x _C − x _D ).

Q Ker θ ⁰ is the radical of Q Σ ⁰ (W _n ).

Σ ⁰ (W n ) ' Mantaci-Reutenauer algebra.

T _n (xy ) = hθ ⁰ (x ), θ ⁰ (y )i.

Mantaci-Reutenauer algebra Definition

x −1,1 x −2 = 2x −1,−1 + 2(x −1,1 − x 1,−1 )

If C is parabolic (i.e. c _i < 0 if i > 2) or if D is almost positive (i.e. d i > −1), then

X _D = a

d∈X

X _C ^C _∩

D .d

Σ(W _n ) ⊂ Σ ⁰ (W _n ) Σ(S _n ) ' ⊕

C ∈Comp

(n) Z x _C ⊂ Σ ⁰ (W _n ) Notation: C ⊂ D ⇔ W _C ⊂ W _D

Σ ⁰ (W D ) = ⊕

C⊂D Z x _C ^D

If D = (d ₁ , . . . , d _r ), then Σ ⁰ (W _D ) ' Σ ⁰ (W _d

) ⊗ _Z · · · ⊗ _Z Σ ⁰ (W _d

) , where Σ ⁰ (S _n ) = Σ(S _n ).

Remark: x _D x _C ^D = x _C .

Mantaci-Reutenauer algebra Definition

x −1,1 x −2 = 2x −1,−1 + 2(x −1,1 − x 1,−1 )

If C is parabolic (i.e. c _i < 0 if i > 2) or if D is almost positive (i.e. d i > −1), then

X _D = a

d∈X

X _C ^C _∩

D .d

Σ(W _n ) ⊂ Σ ⁰ (W _n ) Σ(S _n ) ' ⊕

C ∈Comp

(n) Z x _C ⊂ Σ ⁰ (W _n ) Notation: C ⊂ D ⇔ W _C ⊂ W _D

Σ ⁰ (W D ) = ⊕

C⊂D Z x _C ^D

If D = (d ₁ , . . . , d _r ), then Σ ⁰ (W _D ) ' Σ ⁰ (W _d

) ⊗ _Z · · · ⊗ _Z Σ ⁰ (W _d

) , where Σ ⁰ (S _n ) = Σ(S _n ).

Remark: x _D x _C ^D = x _C .

Mantaci-Reutenauer algebra Definition

x −1,1 x −2 = 2x −1,−1 + 2(x −1,1 − x 1,−1 )

If C is parabolic (i.e. c _i < 0 if i > 2) or if D is almost positive (i.e. d i > −1), then

X _D = a

d∈X

X _C ^C _∩

D .d

Σ(W _n ) ⊂ Σ ⁰ (W _n ) Σ(S _n ) ' ⊕

C ∈Comp

(n) Z x _C ⊂ Σ ⁰ (W _n )

Notation: C ⊂ D ⇔ W _C ⊂ W _D Σ ⁰ (W D ) = ⊕

C⊂D Z x _C ^D

If D = (d ₁ , . . . , d _r ), then Σ ⁰ (W _D ) ' Σ ⁰ (W _d

) ⊗ _Z · · · ⊗ _Z Σ ⁰ (W _d

) , where Σ ⁰ (S _n ) = Σ(S _n ).

Remark: x _D x _C ^D = x _C .

Mantaci-Reutenauer algebra Definition

x −1,1 x −2 = 2x −1,−1 + 2(x −1,1 − x 1,−1 )

If C is parabolic (i.e. c _i < 0 if i > 2) or if D is almost positive (i.e. d i > −1), then

X _D = a

d∈X

X _C ^C _∩

D .d

Σ(W _n ) ⊂ Σ ⁰ (W _n ) Σ(S _n ) ' ⊕

C ∈Comp

(n) Z x _C ⊂ Σ ⁰ (W _n ) Notation: C ⊂ D ⇔ W _C ⊂ W _D

Σ ⁰ (W D ) = ⊕

C⊂D Z x _C ^D

If D = (d ₁ , . . . , d _r ), then Σ ⁰ (W _D ) ' Σ ⁰ (W _d

) ⊗ _Z · · · ⊗ _Z Σ ⁰ (W _d

) , where Σ ⁰ (S _n ) = Σ(S _n ).

Remark: x _D x _C ^D = x _C .

Mantaci-Reutenauer algebra Definition

x −1,1 x −2 = 2x −1,−1 + 2(x −1,1 − x 1,−1 )

If C is parabolic (i.e. c _i < 0 if i > 2) or if D is almost positive (i.e. d i > −1), then

X _D = a

d∈X

X _C ^C _∩

D .d

Σ(W _n ) ⊂ Σ ⁰ (W _n ) Σ(S _n ) ' ⊕

C ∈Comp

(n) Z x _C ⊂ Σ ⁰ (W _n ) Notation: C ⊂ D ⇔ W _C ⊂ W _D

Σ ⁰ (W D ) = ⊕

C⊂D Z x _C ^D

If D = (d ₁ , . . . , d _r ), then Σ ⁰ (W _D ) ' Σ ⁰ (W _d

) ⊗ _Z · · · ⊗ _Z Σ ⁰ (W _d

) , where Σ ⁰ (S _n ) = Σ(S _n ).

Remark: x _D x _C ^D = x _C .

Mantaci-Reutenauer algebra Definition

x −1,1 x −2 = 2x −1,−1 + 2(x −1,1 − x 1,−1 )

If C is parabolic (i.e. c _i < 0 if i > 2) or if D is almost positive (i.e. d i > −1), then

X _D = a

d∈X

X _C ^C _∩

D .d

Σ(W _n ) ⊂ Σ ⁰ (W _n ) Σ(S _n ) ' ⊕

C ∈Comp

(n) Z x _C ⊂ Σ ⁰ (W _n ) Notation: C ⊂ D ⇔ W _C ⊂ W _D

Σ ⁰ (W D ) = ⊕

C⊂D Z x _C ^D

If D = (d ₁ , . . . , d _r ), then Σ ⁰ (W _D ) ' Σ ⁰ (W _d

) ⊗ _Z · · · ⊗ _Z Σ ⁰ (W _d

) , where Σ ⁰ (S _n ) = Σ(S _n ).

Remark: x _D x _C ^D = x _C .

Representation theory in characteristic zero Simples

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Representation theory in characteristic zero Simples

cox _C : Coxeter element of W _C

λ : Comp(n) −→ Bip(n) (Example: λ(3, −1, −3, 1, −2, −2, 4) = (431; 3221))

cox _C ∼ cox _D ⇔ W _C ' W _D ⇔ C ≡ D ⇔ λ(C ) = λ(D) C λ : conjugacy class of cox C if λ(C ) = λ; cox λ ∈ C λ .

Bip(n) − → ^∼ W _n / ∼

λ 7→ C _λ

w =

4 5 11 12

−11 12 −5 −4

Representation theory in characteristic zero Simples

cox _C : Coxeter element of W _C λ : Comp(n) −→ Bip(n)

(Example: λ(3, −1, −3, 1, −2, −2, 4) = (431; 3221))

cox _C ∼ cox _D ⇔ W _C ' W _D ⇔ C ≡ D ⇔ λ(C ) = λ(D) C λ : conjugacy class of cox C if λ(C ) = λ; cox λ ∈ C λ .

Bip(n) − → ^∼ W _n / ∼

λ 7→ C _λ

w =

4 5 11 12

−11 12 −5 −4

Representation theory in characteristic zero Simples

cox _C : Coxeter element of W _C λ : Comp(n) −→ Bip(n) (Example:

λ(3, −1, −3, 1, −2, −2, 4) = (431; 3221))

cox _C ∼ cox _D ⇔ W _C ' W _D ⇔ C ≡ D ⇔ λ(C ) = λ(D) C λ : conjugacy class of cox C if λ(C ) = λ; cox λ ∈ C λ .

Bip(n) − → ^∼ W _n / ∼

λ 7→ C _λ

w =

4 5 11 12

−11 12 −5 −4

Representation theory in characteristic zero Simples

cox _C : Coxeter element of W _C λ : Comp(n) −→ Bip(n) (Example:

λ(3, −1, −3, 1, −2, −2, 4) = (431; 3221))

cox _C ∼ cox _D ⇔ W _C ' W _D ⇔ C ≡ D ⇔ λ(C ) = λ(D) C λ : conjugacy class of cox C if λ(C ) = λ; cox λ ∈ C λ .

Bip(n) − → ^∼ W _n / ∼

λ 7→ C _λ

w =

4 5 11 12

−11 12 −5 −4

Representation theory in characteristic zero Simples

cox _C : Coxeter element of W _C λ : Comp(n) −→ Bip(n) (Example:

λ(3, −1, −3, 1, −2, −2, 4) = (431; 3221))

cox _C ∼ cox _D ⇔ W _C ' W _D ⇔ C ≡ D ⇔ λ(C ) = λ(D) C λ : conjugacy class of cox C if λ(C ) = λ; cox λ ∈ C λ .

Bip(n) − → ^∼ W _n / ∼

λ 7→ C _λ

w =

4 5 11 12

−11 12 −5 −4

Representation theory in characteristic zero Simples

cox C : Coxeter element of W C

λ : Comp(n) −→ Bip(n) (Example:

λ(3, −1, −3, 1, −2, −2, 4) = (431; 3221))

cox _C ∼ cox _D ⇔ W _C ' W _D ⇔ C ≡ D ⇔ λ(C ) = λ(D) C _λ : conjugacy class of cox _C if λ(C ) = λ; cox _λ ∈ C _λ .

Bip(n) − → ^∼ W n / ∼

λ 7→ C _λ

w =

1 2 3 4 5 6 7 8 9 10 11 12 3 − 8 6 −11 12 − 7 − 1 9 2 − 10 −5 −4

Representation theory in characteristic zero Simples

cox C : Coxeter element of W C

λ : Comp(n) −→ Bip(n) (Example:

λ(3, −1, −3, 1, −2, −2, 4) = (431; 3221))

cox _C ∼ cox _D ⇔ W _C ' W _D ⇔ C ≡ D ⇔ λ(C ) = λ(D) C _λ : conjugacy class of cox _C if λ(C ) = λ; cox _λ ∈ C _λ .

Bip(n) − → ^∼ W n / ∼

λ 7→ C _λ

w =

1 2 3 4 5 6 7 8 9 10 11 12 3 −8 6 −11 12 −7 −1 9 2 −10 −5 −4

Representation theory in characteristic zero Simples

cox C : Coxeter element of W C

λ : Comp(n) −→ Bip(n) (Example:

λ(3, −1, −3, 1, −2, −2, 4) = (431; 3221))

cox _C ∼ cox _D ⇔ W _C ' W _D ⇔ C ≡ D ⇔ λ(C ) = λ(D) C _λ : conjugacy class of cox _C if λ(C ) = λ; cox _λ ∈ C _λ .

Bip(n) − → ^∼ W n / ∼

λ 7→ C _λ

w =

1 2 3 4 5 6 7 8 9 10 11 12 3 −8 6 −11 12 −7 −1 9 2 −10 −5 −4

w ∈ C 431;4

Representation theory in characteristic zero Simples

Let

π λ : Σ ⁰ (W n ) −→ Z

x 7−→ θ ⁰ (x)(cox _λ ) This is a morphism of algebras.

If R is a commutative ring, π _λ ^R : RΣ ⁰ (W _n ) → R and D ^R _λ is the R Σ ⁰ (W _n )-module which is R -free of rank 1 and on which R Σ ⁰ (W _n ) acts through π ^R _λ .

Irr Q Σ ⁰ (W _n ) = {π ^Q _λ | λ ∈ Bip(n)}

Example: character table of Q Σ ⁰ (W ₂ )

x ₂ x ¯ 2 x _1,1 x _1, ¯ 1 x ¯ 1, ¯ 1

π _2;∅ ^Q 1 . . . .

π _∅ ^Q _;2 1 2 . . .

π _11; ^Q

∅ 1 . 2 . .

π _1;1 ^Q 1 . 2 2 .

π _∅;11 ^Q 1 4 2 4 8

Representation theory in characteristic zero Simples

Let

π λ : Σ ⁰ (W n ) −→ Z

x 7−→ θ ⁰ (x)(cox _λ ) This is a morphism of algebras.

If R is a commutative ring, π _λ ^R : RΣ ⁰ (W _n ) → R and D ^R _λ is the RΣ ⁰ (W _n )-module which is R -free of rank 1 and on which RΣ ⁰ (W _n ) acts through π ^R _λ .

Irr Q Σ ⁰ (W _n ) = {π ^Q _λ | λ ∈ Bip(n)}

Example: character table of Q Σ ⁰ (W ₂ )

x ₂ x ¯ 2 x _1,1 x _1, ¯ 1 x ¯ 1, ¯ 1

π _2;∅ ^Q 1 . . . .

π _∅ ^Q _;2 1 2 . . .

π _11; ^Q

∅ 1 . 2 . .

π _1;1 ^Q 1 . 2 2 .

π _∅;11 ^Q 1 4 2 4 8

Representation theory in characteristic zero Simples

Let

π λ : Σ ⁰ (W n ) −→ Z

x 7−→ θ ⁰ (x)(cox _λ ) This is a morphism of algebras.

If R is a commutative ring, π _λ ^R : RΣ ⁰ (W _n ) → R and D ^R _λ is the RΣ ⁰ (W _n )-module which is R -free of rank 1 and on which RΣ ⁰ (W _n ) acts through π ^R _λ .

Irr Q Σ ⁰ (W _n ) = {π ^Q _λ | λ ∈ Bip(n)}

Example: character table of Q Σ ⁰ (W ₂ )

x ₂ x ¯ 2 x _1,1 x _1, ¯ 1 x ¯ 1, ¯ 1

π _2;∅ ^Q 1 . . . .

π _∅ ^Q _;2 1 2 . . .

π _11; ^Q

∅ 1 . 2 . .

π _1;1 ^Q 1 . 2 2 .

π _∅;11 ^Q 1 4 2 4 8

Representation theory in characteristic zero Simples

Let

π λ : Σ ⁰ (W n ) −→ Z

x 7−→ θ ⁰ (x)(cox _λ ) This is a morphism of algebras.

If R is a commutative ring, π _λ ^R : RΣ ⁰ (W _n ) → R and D ^R _λ is the RΣ ⁰ (W _n )-module which is R -free of rank 1 and on which RΣ ⁰ (W _n ) acts through π ^R _λ .

Irr Q Σ ⁰ (W _n ) = {π ^Q _λ | λ ∈ Bip(n)}

Example: character table of Q Σ ⁰ (W ₂ )

x ₂ x ¯ 2 x _1,1 x _1, ¯ 1 x ¯ 1, ¯ 1

π _2;∅ ^Q 1 . . . .

π _∅ ^Q _;2 1 2 . . .

π _11; ^Q

∅ 1 . 2 . .

π _1;1 ^Q 1 . 2 2 .

π _∅;11 ^Q 1 4 2 4 8

Representation theory in characteristic zero Simples

Let

π λ : Σ ⁰ (W n ) −→ Z

x 7−→ θ ⁰ (x)(cox _λ ) This is a morphism of algebras.

If R is a commutative ring, π _λ ^R : RΣ ⁰ (W _n ) → R and D ^R _λ is the RΣ ⁰ (W _n )-module which is R -free of rank 1 and on which RΣ ⁰ (W _n ) acts through π ^R _λ .

Irr Q Σ ⁰ (W _n ) = {π ^Q _λ | λ ∈ Bip(n)}

Example: character table of Q Σ ⁰ (W ₂ )

x 2 x ¯ 2 x 1,1 x _1, ¯ 1 x ¯ 1, ¯ 1

π _2;∅ ^Q 1 . . . .

π _∅ ^Q _;2 1 2 . . .

π _11; ^Q

∅ 1 . 2 . .

π _1;1 ^Q 1 . 2 2 .

π _∅;11 ^Q 1 4 2 4 8

Representation theory in characteristic zero Projectives, Cartan matrix

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Representation theory in characteristic zero Projectives, Cartan matrix

x _n = 1 = X

λ∈Bip(n)

E _λ

(E _λ E _µ = δ _λµ E _λ )

such that P _λ ^Q := Q Σ ⁰ (W _n )E _λ is the projective cover of D ^Q _λ

Note that

θ ⁰ (E _λ ) is the characteristic function of C _λ Moreover,

dim _Q Q W _n E _λ = |C _λ |

(Indeed, dim _Q Q W n E λ = |W n |T n (E λ ) = |W n |hθ ⁰ (E λ ), θ ⁰ (1)i) Question: Q W _n E _λ ' Ind ^W _C

Wn

(cox

) ξ _λ for some linear character ξ _λ of

C _W

(cox _λ )? (true for n 6 5)

Representation theory in characteristic zero Projectives, Cartan matrix

x _n = 1 = X

λ∈Bip(n)

E _λ

(E _λ E _µ = δ _λµ E _λ )

such that P _λ ^Q := Q Σ ⁰ (W _n )E _λ is the projective cover of D ^Q _λ Note that

θ ⁰ (E _λ ) is the characteristic function of C _λ

Moreover,

dim _Q Q W _n E _λ = |C _λ |

(Indeed, dim _Q Q W n E λ = |W n |T n (E λ ) = |W n |hθ ⁰ (E λ ), θ ⁰ (1)i) Question: Q W _n E _λ ' Ind ^W _C

Wn

(cox

) ξ _λ for some linear character ξ _λ of

C _W

(cox _λ )? (true for n 6 5)

Representation theory in characteristic zero Projectives, Cartan matrix

x _n = 1 = X

λ∈Bip(n)

E _λ

(E _λ E _µ = δ _λµ E _λ )

such that P _λ ^Q := Q Σ ⁰ (W _n )E _λ is the projective cover of D ^Q _λ Note that

θ ⁰ (E _λ ) is the characteristic function of C _λ Moreover,

dim _Q Q W n E λ = |C λ |

(Indeed, dim _Q Q W n E λ = |W n |T n (E λ ) = |W n |hθ ⁰ (E λ ), θ ⁰ (1)i)

Question: Q W _n E _λ ' Ind ^W _C

Wn

(cox

) ξ _λ for some linear character ξ _λ of

C _W

(cox _λ )? (true for n 6 5)

Representation theory in characteristic zero Projectives, Cartan matrix

x _n = 1 = X

λ∈Bip(n)

E _λ

(E _λ E _µ = δ _λµ E _λ )

such that P _λ ^Q := Q Σ ⁰ (W _n )E _λ is the projective cover of D ^Q _λ Note that

θ ⁰ (E _λ ) is the characteristic function of C _λ Moreover,

dim _Q Q W n E λ = |C λ |

(Indeed, dim _Q Q W n E λ = |W n |T n (E λ ) = |W n |hθ ⁰ (E λ ), θ ⁰ (1)i) Question: Q W _n E _λ ' Ind ^W _C

Wn

(cox

) ξ _λ for some linear character ξ _λ of

C _W

(cox _λ )? (true for n 6 5)

Representation theory in characteristic zero Projectives, Cartan matrix

D _3; ^Q

∅ D _21; ^Q

∅ D ^Q

∅ ;21 D _1;1 ^Q

D ^Q

∅ ;3 D _2;1 ^Q D _1;2 ^Q D ₁ ^Q

;1 D ₁ ^Q

; ∅ D ^Q

∅ ;1

P _3; ^Q _∅ 1 1 1 1 . . . . . . P _21; ^Q

∅ . 1 . . . . . . . .

P _∅;21 ^Q . . 1 1 . . . . . . P _1;1 ^Q

. . . 1 . . . . . . P ^Q

∅ ;3 . . . . 1 1 1 1 . .

P _2;1 ^Q . . . . . 1 . . . .

P _1;2 ^Q . . . . . . 1 1 . .

P ₁ ^Q

;1 . . . . . . . 1 . .

P ₁ ^Q

; ∅ . . . . . . . . 1 .

P _∅;1 ^Q

. . . . . . . . . 1

Representation theory in characteristic zero Projectives, Cartan matrix

4 31 ∅ ∅ 21² 2 1 1² ∅ 3 1 2 21 1² ∅ 1 1³ 2² 1⁴ ∅

∅ ∅ 31 2² ∅ 1² 21 1² 4 1 3 2 1 2 21² 1³ 1 ∅ ∅ 1⁴

4;∅ 1 1 1 . 1 1 2 1 . . . .

31;∅ . 1 . . 1 . 1 1 . . . .

∅; 31 . . 1 . . 1 1 1 . . . .

∅; 2² . . . 1 . . 1 1 . . . .

21²;∅ . . . . 1 . . . .

2; 1² . . . 1 . . . .

1; 21 . . . 1 1 . . . .

1²; 1² . . . 1 . . . .

∅; 4 . . . 1 1 1 1 2 1 1 1 1 . . .

3; 1 . . . 1 . . 1 . 1 1 . . . .

1; 3 . . . 1 . 1 1 . . 1 . . .

2; 2 . . . 1 1 . . . .

21; 1 . . . 1 . . . .

1²; 2 . . . 1 . . 1 . . .

∅; 21² . . . 1 1 . . . .

1; 1³ . . . 1 . . . .

1³; 1 . . . 1 . . .

2²;∅ . . . 1 . .

1⁴;∅ . . . 1 .

∅; 1⁴ . . . 1

Representation theory in characteristic zero Projectives, Cartan matrix

Theorem (B., 2005). Let λ, µ ∈ Bip(n).

[P _λ ^Q : D ^Q _λ ] = 1

If λ 6= µ and [P _λ ^Q : D _µ ^Q ] 6= 0, then

I

lg(λ) > lg(µ)

I

lg(λ ⁻ ) ≡ lg(µ ⁻ ) mod 2

(Note that π λ (w 0 ) = (−1) ^n−lg(λ

⁾ )

Representation theory in characteristic zero Projectives, Cartan matrix

Theorem (B., 2005). Let λ, µ ∈ Bip(n).

[P _λ ^Q : D ^Q _λ ] = 1

If λ 6= µ and [P _λ ^Q : D _µ ^Q ] 6= 0, then

I

lg(λ) > lg(µ)

I

lg(λ ⁻ ) ≡ lg(µ ⁻ ) mod 2

(Note that π λ (w 0 ) = (−1) ^n−lg(λ

⁾ )

Representation theory in characteristic zero Restriction morphisms

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Representation theory in characteristic zero Restriction morphisms

Let D be almost positive and let Res D x C := X

d∈X

x

^D

C ∩D ∈ Σ ⁰ (W D ) = ⊕

C ⊂D Z x _C ^D

Then xx D = x D Res D (x ).

(Note that the map Z W _D → Z W _n , a 7→ x _D a is injective)

Theorem (B., 2005).

Res _D is a morphism of algebras. θ ⁰ _D ◦ Res D = Res ^W _W

D

◦θ ⁰

Q Σ ⁰ (W _n ) = Ker(Res ^Q _D ) ⊕ Q Σ ⁰ (W _n )x _D

Representation theory in characteristic zero Restriction morphisms

Let D be almost positive and let Res D x C := X

d∈X

x

^D

C ∩D ∈ Σ ⁰ (W D ) = ⊕

C ⊂D Z x _C ^D Then xx D = x D Res D (x).

(Note that the map Z W _D → Z W _n , a 7→ x _D a is injective)

Theorem (B., 2005).

Res _D is a morphism of algebras. θ ⁰ _D ◦ Res D = Res ^W _W

D

◦θ ⁰

Q Σ ⁰ (W _n ) = Ker(Res ^Q _D ) ⊕ Q Σ ⁰ (W _n )x _D

Representation theory in characteristic zero Restriction morphisms

Let D be almost positive and let Res D x C := X

d∈X

x

^D

C ∩D ∈ Σ ⁰ (W D ) = ⊕

C ⊂D Z x _C ^D Then xx D = x D Res D (x).

(Note that the map Z W _D → Z W _n , a 7→ x _D a is injective)

Theorem (B., 2005).

Res _D is a morphism of algebras.

θ ⁰ _D ◦ Res D = Res ^W _W

D

◦θ ⁰

Q Σ ⁰ (W _n ) = Ker(Res ^Q _D ) ⊕ Q Σ ⁰ (W _n )x _D

Representation theory in characteristic zero Restriction morphisms

4 31 ∅ ∅ 21² 2 1 1² ∅ 3 1 2 21 1² ∅ 1 1³ 2² 1⁴ ∅

∅ ∅ 31 2² ∅ 1² 21 1² 4 1 3 2 1 2 21² 1³ 1 ∅ ∅ 1⁴

4;∅ 1 1 1 . 1 1 2 1 . . . .

31;∅ . 1 . . 1 . 1 1 . . . .

∅; 31 . . 1 . . 1 1 1 . . . .

∅; 2² . . . 1 . . 1 1 . . . .

21²;∅ . . . . 1 . . . .

2; 1² . . . 1 . . . .

1; 21 . . . 1 1 . . . . 1²; 1² . . . 1 . . . .

∅; 4 . . . 1 1 1 1 2 1 1 1 1 . . . 3; 1 . . . 1 . . 1 . 1 1 . . . . 1; 3 . . . 1 . 1 1 . . 1 . . .

2; 2 . . . 1 1 . . . . 21; 1 . . . 1 . . . .

1²; 2 . . . 1 . . 1 . . .

∅; 21² . . . 1 1 . . . . 1; 1³ . . . 1 . . . .

1³; 1 . . . 1 . . .

2²;∅ . . . 1 . .

1⁴;∅ . . . 1 .

∅; 1⁴ . . . 1

Representation theory in characteristic zero Restriction morphisms

4 31 ∅ ∅ 21² 2 1 1² ∅ 3 1 2 21 1² ∅ 1 1³ 2² 1⁴ ∅

∅ ∅ 31 2² ∅ 1² 21 1² 4 1 3 2 1 2 21² 1³ 1 ∅ ∅ 1⁴

4;∅ 1 1 1 . 1 1 2 1 . . . .

31;∅ . 1 . . 1 . 1 1 . . . .

∅; 31 . . 1 . . 1 1 1 . . . .

∅; 2² . . . 1 . . 1 1 . . . .

21²;∅ . . . . 1 . . . .

2; 1² . . . 1 . . . .

1; 21 . . . 1 1 . . . . 1²; 1² . . . 1 . . . .

∅; 4 . . . 1 1 1 1 2 1 1 1 1 . . . 3; 1 . . . 1 . . 1 . 1 1 . . . . 1; 3 . . . 1 . 1 1 . . 1 . . .

2; 2 . . . 1 1 . . . . 21; 1 . . . 1 . . . .

1²; 2 . . . 1 . . 1 . . .

∅; 21² . . . 1 1 . . . . 1; 1³ . . . 1 . . . .

1³; 1 . . . 1 . . .

2²;∅ . . . 1 . .

1⁴;∅ . . . 1 .

∅; 1⁴ . . . 1

Representation theory in characteristic zero Restriction morphisms

D _3; ^Q

∅ D _21; ^Q

∅ D ^Q

∅ ;21 D _1;1 ^Q

D ^Q

∅ ;3 D _2;1 ^Q D _1;2 ^Q D ₁ ^Q

;1 D ₁ ^Q

; ∅ D ^Q

∅ ;1

P _3; ^Q _∅ 1 1 1 1 . . . . . . P _21; ^Q

∅ . 1 . . . . . . . .

P _∅;21 ^Q . . 1 1 . . . . . . P _1;1 ^Q

. . . 1 . . . . . . P ^Q

∅ ;3 . . . . 1 1 1 1 . .

P _2;1 ^Q . . . . . 1 . . . .

P _1;2 ^Q . . . . . . 1 1 . .

P ₁ ^Q

;1 . . . . . . . 1 . .

P ₁ ^Q

; ∅ . . . . . . . . 1 .

P _∅;1 ^Q

. . . . . . . . . 1

Representation theory in characteristic zero Restriction morphisms

The natural map W n−1 , → W n induces an injective map τ _n : Bip(n − 1) → Bip(n). In fact:

τ _n ((λ ⁺ ₁ , . . . , λ ⁺ _r ), (λ ⁻ ₁ , . . . , λ ⁻ _s )) = ((λ ⁺ ₁ , . . . , λ ⁺ _r ), (λ ⁻ ₁ , . . . , λ ⁻ _s , 1)).

Theorem (B., 2006).

Res _n−1,−1 : Q Σ ⁰ (W _n ) → Q Σ ⁰ (W _n−1 ) is surjective.

It is probable that Res n−1,−1 : Σ ⁰ (W _n ) → Σ ⁰ (W n−1 ) is also surjective (this is true for n 6 5). Note that Res ^W _W

: Z Irr W _n → Z Irr W n−1

is surjective.

”Corollary”. If λ, µ ∈ Bip(n − 1), then [P _λ ^Q , D ^Q _µ ] = [P _τ

_(λ) , D ^Q _τ

n

(µ) ].

Representation theory in characteristic zero Restriction morphisms

The natural map W n−1 , → W n induces an injective map τ _n : Bip(n − 1) → Bip(n). In fact:

τ _n ((λ ⁺ ₁ , . . . , λ ⁺ _r ), (λ ⁻ ₁ , . . . , λ ⁻ _s )) = ((λ ⁺ ₁ , . . . , λ ⁺ _r ), (λ ⁻ ₁ , . . . , λ ⁻ _s , 1)).

Theorem (B., 2006).

Res _n−1,−1 : Q Σ ⁰ (W _n ) → Q Σ ⁰ (W _n−1 ) is surjective.

It is probable that Res n−1,−1 : Σ ⁰ (W _n ) → Σ ⁰ (W n−1 ) is also surjective (this is true for n 6 5). Note that Res ^W _W

: Z Irr W _n → Z Irr W n−1

is surjective.

”Corollary”. If λ, µ ∈ Bip(n − 1), then [P _λ ^Q , D ^Q _µ ] = [P _τ

_(λ) , D ^Q _τ

n

(µ) ].

Representation theory in characteristic zero Restriction morphisms

The natural map W n−1 , → W n induces an injective map τ _n : Bip(n − 1) → Bip(n). In fact:

τ _n ((λ ⁺ ₁ , . . . , λ ⁺ _r ), (λ ⁻ ₁ , . . . , λ ⁻ _s )) = ((λ ⁺ ₁ , . . . , λ ⁺ _r ), (λ ⁻ ₁ , . . . , λ ⁻ _s , 1)).

Theorem (B., 2006).

Res _n−1,−1 : Q Σ ⁰ (W _n ) → Q Σ ⁰ (W _n−1 ) is surjective.

It is probable that Res n−1,−1 : Σ ⁰ (W _n ) → Σ ⁰ (W n−1 ) is also surjective (this is true for n 6 5). Note that Res ^W _W

: Z Irr W _n → Z Irr W n−1

is surjective.

”Corollary”. If λ, µ ∈ Bip(n − 1), then [P _λ ^Q , D ^Q _µ ] = [P _τ

_(λ) , D ^Q _τ

n

(µ) ].

Representation theory in characteristic zero Restriction morphisms

The natural map W n−1 , → W n induces an injective map τ _n : Bip(n − 1) → Bip(n). In fact:

τ _n ((λ ⁺ ₁ , . . . , λ ⁺ _r ), (λ ⁻ ₁ , . . . , λ ⁻ _s )) = ((λ ⁺ ₁ , . . . , λ ⁺ _r ), (λ ⁻ ₁ , . . . , λ ⁻ _s , 1)).

Theorem (B., 2006).

Res _n−1,−1 : Q Σ ⁰ (W _n ) → Q Σ ⁰ (W _n−1 ) is surjective.

It is probable that Res n−1,−1 : Σ ⁰ (W _n ) → Σ ⁰ (W n−1 ) is also surjective (this is true for n 6 5). Note that Res ^W _W

: Z Irr W _n → Z Irr W n−1

is surjective.

”Corollary”. If λ, µ ∈ Bip(n − 1), then [P _λ ^Q , D ^Q _µ ] = [P _τ

_(λ) , D ^Q _τ

n

(µ) ].

Modular representation theory Simples, radical

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Modular representation theory Simples, radical

Let Bip _p

(n) be the set of bipartitions λ such that cox λ is p-regular.

If λ ∈ Bip(n), let λ _p

denote the unique bipartition in Bip _p

(n) such that cox _λ

is conjugate to the p ⁰ -part of cox _λ .

Let Comp _p (n) := {C ∈ Comp(n) | p divides |N _W

(W _C )/W _C |}.

Theorem (B., 2005). π ^F _λ

= π µ ^F

⇔ λ _p

= µ _p

Irr F ^p Σ ⁰ (W _n ) = {π _λ ^F

| λ ∈ Bip _p

(n)} Rad F p Σ ⁰ (W _n ) = X

C≡D

F p (x _C − x _D ) + X

C ∈Comp

(n)

F p x _C

Modular representation theory Simples, radical

Let Bip _p

(n) be the set of bipartitions λ such that cox λ is p-regular.

If λ ∈ Bip(n), let λ _p

denote the unique bipartition in Bip _p

(n) such that cox _λ

is conjugate to the p ⁰ -part of cox _λ .

Let Comp _p (n) := {C ∈ Comp(n) | p divides |N _W

(W _C )/W _C |}.

Theorem (B., 2005). π ^F _λ

= π µ ^F

⇔ λ _p

= µ _p

Irr F ^p Σ ⁰ (W _n ) = {π _λ ^F

| λ ∈ Bip _p

(n)} Rad F p Σ ⁰ (W _n ) = X

C≡D

F p (x _C − x _D ) + X

C ∈Comp

(n)

F p x _C

Modular representation theory Simples, radical

Let Bip _p

(n) be the set of bipartitions λ such that cox λ is p-regular.

If λ ∈ Bip(n), let λ _p

denote the unique bipartition in Bip _p

(n) such that cox _λ

is conjugate to the p ⁰ -part of cox _λ .

Let Comp _p (n) := {C ∈ Comp(n) | p divides |N _W

(W _C )/W _C |}.

Theorem (B., 2005). π ^F _λ

= π µ ^F

⇔ λ _p

= µ _p

Irr F ^p Σ ⁰ (W _n ) = {π _λ ^F

| λ ∈ Bip _p

(n)} Rad F p Σ ⁰ (W _n ) = X

C≡D

F p (x _C − x _D ) + X

C ∈Comp

(n)

F p x _C

Modular representation theory Simples, radical

Let Bip _p

(n) be the set of bipartitions λ such that cox λ is p-regular.

If λ ∈ Bip(n), let λ _p

denote the unique bipartition in Bip _p

(n) such that cox _λ

is conjugate to the p ⁰ -part of cox _λ .

Let Comp _p (n) := {C ∈ Comp(n) | p divides |N _W

(W _C )/W _C |}.

Theorem (B., 2005).

π ^F _λ

= π µ ^F

⇔ λ _p

= µ _p

Irr F ^p Σ ⁰ (W _n ) = {π _λ ^F

| λ ∈ Bip _p

(n)}

Rad F p Σ ⁰ (W _n ) = X

C≡D

F p (x _C − x _D ) + X

C ∈Comp

(n)

F p x _C

Modular representation theory Cartan matrix

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Modular representation theory Cartan matrix

Let ∆ _n (p) denote the matrix (δ _λ

,µ ) λ∈Bip(n),µ∈Bip

(n) . This is the decomposition matrix from Q Σ ⁰ (W n ) to F ^p Σ ⁰ (W n ).

By a general result of Geck and Rouquier, we have

Cartan( F p Σ ⁰ (W _n )) = ^t ∆ _n (p) × Cartan( Q Σ ⁰ (W _n )) × ∆ _n (p)

Modular representation theory Cartan matrix

Let ∆ _n (p) denote the matrix (δ _λ

,µ ) λ∈Bip(n),µ∈Bip

(n) . This is the decomposition matrix from Q Σ ⁰ (W n ) to F ^p Σ ⁰ (W n ).

By a general result of Geck and Rouquier, we have

Cartan( F p Σ ⁰ (W _n )) = ^t ∆ _n (p) × Cartan( Q Σ ⁰ (W _n )) × ∆ _n (p)

Loewy length Character ring

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Loewy length Character ring

Theorem (B., 2006). Let K be a field of characteristic p > 0.

Then the Loewy length of K Irr W _n is equal to



 

 

1, if p = 0;

n + 1, if p = 2;

[n/p] + 1, if p > 2.

Loewy length Mantaci-Reutenauer algebra

Contents

1 Solomon’s descent algebra

2 Mantaci-Reutenauer algebra Coxeter group of type B Some reflection subgroups Definition

3 Representation theory in characteristic zero Simples

Projectives, Cartan matrix Restriction morphisms

4 Modular representation theory Simples, radical

Cartan matrix

5 Loewy length Character ring

Mantaci-Reutenauer algebra

Loewy length Mantaci-Reutenauer algebra

Theorem (B., 2006). Let K be a field of characteristic p > 0.

Then the Loewy length of K Σ ⁰ (W _n ) is equal to



 

 

n, if p 6= 2;

2, if p = 2 and n = 1;

2n − 1 ?, if p = 2 and n > 2.