Families of characters for cyclotomic Hecke algebras

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Families of characters for cyclotomic Hecke algebras

Maria Chlouveraki EPFL 30 MAIOY 2008

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A complex reflection groupW is a finite group of matrices with coefficients in a number field K generated by pseudo-reflections,i.e., elements whose vector space of fixed points is a hyperplane.

IfK =Q, then W is a Weyl group.

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A complex reflection groupW is a finite group of matrices with coefficients in a number field K generated by pseudo-reflections,i.e., elements whose vector space of fixed points is a hyperplane.

IfK =Q, then W is a Weyl group.

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Weyl groups →

Complex reflection groups Finite reductive groups → “Spetses” (?)

Families of characters → Rouquier blocks

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Weyl groups → Complex reflection groups

Finite reductive groups → “Spetses” (?) Families of characters → Rouquier blocks

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Weyl groups → Complex reflection groups Finite reductive groups →

“Spetses” (?) Families of characters → Rouquier blocks

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Weyl groups → Complex reflection groups Finite reductive groups → “Spetses” (?)

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Families of characters →

Rouquier blocks

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Hecke algebras of complex reflection groups

Every complex reflection group W has a nice“presentation a la Coxeter”:

G2 = <s,t|(st)³= (ts)³, s² =t² = 1>

= <s,t|(st)³= (ts)³,(s −1)(s+ 1) = (t−1)(t+ 1) = 0>

The generic Hecke algebraH(W) has a presentation of the form:

H(G₂) =< σ, τ|(στ)³ = (τ σ)³,(σ−u₀)(σ−u₁) = (τ−u₂)(τ−u₃) = 0>

and it’s defined over the Laurent polynomial ring Z[u,u⁻¹], where u= (u0,u1,u2,u3) is a set of indeterminates.

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Hecke algebras of complex reflection groups

G2 = <s,t|(st)³= (ts)³, s² =t² = 1>

= <s,t|(st)³= (ts)³,(s−1)(s+ 1) = (t−1)(t+ 1) = 0>

and it’s defined over the Laurent polynomial ring Z[u,u⁻¹], where u= (u0,u1,u2,u3) is a set of indeterminates.

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Hecke algebras of complex reflection groups

G2 = <s,t|(st)³= (ts)³, s² =t² = 1>

= <s,t|(st)³= (ts)³,(s−1)(s+ 1) = (t−1)(t+ 1) = 0>

and it’s defined over the Laurent polynomial ring Z[u,u⁻¹], where

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A theorem by G. Malle provides us with a set of indeterminates vsuch that the the K(v)-algebra K(v)H(W) is split semisimple:

v₀² =u₀, v₁²=−u₁, v₂²=u₂, v₃² =−u₃

By “Tits’ deformation theorem”, the specialization v_j 7→1 induces a bijection

Irr(K(v)H(W)) ↔ Irr(W)

χ_v 7→ χ

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A theorem by G. Malle provides us with a set of indeterminates vsuch that the the K(v)-algebra K(v)H(W) is split semisimple:

v₀² =u₀, v₁²=−u₁, v₂²=u₂, v₃² =−u₃

By “Tits’ deformation theorem”, the specialization vj 7→1 induces a bijection

Irr(K(v)H(W)) ↔ Irr(W)

χ_v 7→ χ

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Generic Schur elements

The generic Hecke algebra is endowed with a canonical symmetrizing form t. We have that

t = X

χ∈Irr(W)

1 sχ

χv,

where sχ is the Schur element associated toχv∈Irr(K(v)H(W)). Theorem (C.)

The generic Schur elements are polynomials inZK[v,v⁻¹] whose irreducible factors are of the form Ψ(M),

where Ψis aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0,

i.e.,if M =Q

jv_j^a^j, thengcd(a_j) = 1 and P

ja_j = 0.

The primitive monomials appearing in the factorization of sχ are unique up to inversion.

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Generic Schur elements

t = X

χ∈Irr(W)

1 s_χχv,

where sχ is the Schur element associated toχv∈Irr(K(v)H(W)).

Theorem (C.)

The generic Schur elements are polynomials inZK[v,v⁻¹] whose irreducible factors are of the form Ψ(M),

where Ψis aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0,

i.e.,if M =Q

ja_j = 0.

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Generic Schur elements

t = X

χ∈Irr(W)

1 s_χχv,

Theorem (C.)

The generic Schur elements are polynomials inZK[v,v⁻¹] whose irreducible factors are of the form Ψ(M), where

Ψis aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0,

i.e.,if M =Q

ja_j = 0.

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Generic Schur elements

t = X

χ∈Irr(W)

1 s_χχv,

Theorem (C.)

i.e.,if M =Q

jv_j^a^j, thengcd(aj) = 1 andP

jaj = 0.

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Generic Schur elements

t = X

χ∈Irr(W)

1 s_χχv,

Theorem (C.)

i.e.,if M =Q

jv_j^a^j, thengcd(aj) = 1 andP

jaj = 0.

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Schur elements of G2

s1 = Φ4(v0v₁⁻¹)·Φ4(v2v₃⁻¹)·Φ3(v0v₁⁻¹v2v₃⁻¹)·Φ6(v0v₁⁻¹v2v₃⁻¹) s₂ = 2·v₁²v₀⁻²·Φ₃(v₀v₁⁻¹v₂v₃⁻¹)·Φ₆(v₀v₁⁻¹v₂⁻¹v₃)

Φ₄(x) =x²+ 1, Φ₃(x) =x²+x+ 1, Φ₆(x) =x²−x+ 1.

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Cyclotomic Hecke algebras

Definition

Let y be an indeterminate. A cyclotomic specializationofH(W) is a ZK-algebra morphism φ:ZK[v,v⁻¹]→ZK[y,y⁻¹] such that

φ:v_j 7→yⁿ^j, with n_j ∈Zfor all j.

The correspondingcyclotomic Hecke algebra H_φ is theZK[y,y⁻¹]-algebra obtained as the specialization of H(W) via the morphism φ.

Proposition (C.)

The algebra K(y)H_φ is split semisimple.

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Cyclotomic Hecke algebras

Definition

Let y be an indeterminate. Acyclotomic specialization ofH(W) is a ZK-algebra morphism φ:ZK[v,v⁻¹]→ZK[y,y⁻¹] such that

The correspondingcyclotomic Hecke algebra H_φis the ZK[y,y⁻¹]-algebra obtained as the specialization of H(W) via the morphism φ.

Proposition (C.)

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Cyclotomic Hecke algebras

Definition

Proposition (C.)

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Cyclotomic Hecke algebras

Definition

Proposition (C.)

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By “Tits’ deformation theorem”, we obtain

Irr(K(v)H(W)) ↔ Irr(K(y)H_φ) ↔ Irr(W)

χv 7→ χφ 7→ χ

Proposition

The Schur element sχφ(y)associated to the irreducible character χφ of K(y)H_φis a Laurent polynomial iny of the form

s_χ_φ(y) =ψ_χ_φy^a^χφ Y

Φ∈C_K

Φ(y)ⁿ^χφ,^Φ,

where ψ_χ_φ ∈ZK,a_χ_φ ∈Z,n_χ_φ_,Φ ∈Nand C_K is a set ofK-cyclotomic polynomials.

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By “Tits’ deformation theorem”, we obtain

Irr(K(v)H(W)) ↔ Irr(K(y)H_φ) ↔ Irr(W)

χv 7→ χφ 7→ χ

Proposition

The Schur element sχφ(y)associated to the irreducible character χφ of K(y)H_φis a Laurent polynomial iny of the form

s_χ_φ(y) =ψ_χ_φy^a^χφ Y

Φ∈CK

Φ(y)ⁿ^χφ,^Φ,

where ψ_χ_φ ∈ZK,a_χ_φ ∈Z,n_χ_φ_,Φ ∈Nand C_K is a set ofK-cyclotomic polynomials.

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Rouquier blocks

We call Rouquier ringof K theZK-subalgebra ofK(y) R_K(y):=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

Definition

The Rouquier blocksof the cyclotomic Hecke algebra H_φ are the blocks of R_K(y)H_φ,

i.e., the partitionBR(H_φ) ofIrr(W) minimal for the property: For allB∈ BR(H_φ)and h∈ H_φ,X

χ∈B

χ_φ(h)

s_χ_φ ∈ R_K(y).

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Rouquier blocks

Definition

i.e., the partitionBR(H_φ) ofIrr(W) minimal for the property: For allB∈ BR(H_φ)and h∈ H_φ,X

χ∈B

χ_φ(h)

s_χ_φ ∈ R_K(y).

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Rouquier blocks

Definition

i.e., the partitionBR(H_φ) ofIrr(W) minimal for the property: For allB∈ BR(H_φ) and h∈ H_φ,X

χ∈B

χ_φ(h)

s_χ_φ ∈ R_K(y).

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Rouquier blocks

Definition

The Rouquier blocksof the cyclotomic Hecke algebra H_φ are the blocks of R_K(y)H_φ,i.e., the partitionBR(H_φ) ofIrr(W) minimal for the property:

For all B∈ BR(H_φ) and h∈ H_φ,X

χ∈B

χ_φ(h)

s_χ_φ ∈ R_K(y).

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Let p be a prime ideal ofZK.

We denote by B_p(H_φ) the partition of Irr(W) into p-blocks ofH_φ (i.e.,the blocks of the algebra ZK[y,y⁻¹]pH_φ).

Proposition

The Rouquier blocks of H_φis the partition of Irr(W) generated by the partitions B_p(H_φ), where pruns over the set of prime ideals of ZK.

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Proposition

The Rouquier blocks of H_φis the partition of Irr(W) generated by the partitions B_p(H_φ), where pruns over the set of prime ideals of ZK.

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Proposition

The Rouquier blocks of H_φ is the partition of Irr(W) generated by the partitions B_p(H_φ), where pruns over the set of prime ideals of ZK.

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p-essential monomials

Definition

A primitive monomial M in ZK[v,v⁻¹] is calledp-essential forW if there exists an irreducible characterχ of W and a K-cyclotomic polynomial Ψ such that

1 Ψ(M) dividessχ(v)

2 Ψ(1)∈p.

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p-essential monomials

Definition

A primitive monomial M in ZK[v,v⁻¹] is calledp-essential for W if there exists an irreducible characterχ of W and a K-cyclotomic polynomial Ψ such that

2 Ψ(1)∈p.

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p-essential monomials

Definition

2 Ψ(1)∈p.

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p-essential monomials

Definition

2 Ψ(1)∈p.

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p-essential monomials and p-blocks

Set A:=ZK[v,v⁻¹] and letB_p(H)be the partition of Irr(W) into p-blocks ofH(W) (i.e., the blocks of the algebra ApH(W)).

Theorem (C.)

For every p-essential monomialM for W, there exists a unique partition B_p^M(H) ofIrr(W) with the following properties:

1 The parts ofB^M_p (H) are unions of the parts ofB_p(H).

2 The partition B_p(H_φ) is the partition generated by the partitions B_p(H) et B_p^M(H), where M runs over the set of allp-essential monomials which are sent to 1 byφ.

Moreover, the partition B^M_p (H) coincides with the blocks of the algebra Aq_MH(W), where q_M := (M −1)A+pA.

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p-essential monomials and p-blocks

Set A:=ZK[v,v⁻¹] and letB_p(H)be the partition of Irr(W) into p-blocks ofH(W) (i.e., the blocks of the algebraApH(W)).

Theorem (C.)

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p-essential monomials and p-blocks

Theorem (C.)

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p-essential monomials and p-blocks

Theorem (C.)

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p-essential monomials and p-blocks

Theorem (C.)

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p-essential monomials and p-blocks

Theorem (C.)

Moreover, the partition B^M_p (H) coincides with the blocks of the algebra A_q_MH(W), where q_M := (M −1)A+pA.

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The example of G

2

We denote the characters of G2 as follows:

χ_1,0, χ_1,6, χ_1,3⁰, χ_1,3⁰⁰, χ_2,1, χ_2,2.

Schur elements: 2-essential in purple, 3-essential in green

s₁ = Φ₄(v₀v₁⁻¹)·Φ₄(v₂v₃⁻¹)·Φ₃(v₀v₁⁻¹v₂v₃⁻¹)·Φ₆(v₀v₁⁻¹v₂v₃⁻¹)

s2 = 2·v₁²v₀⁻²·Φ3(v0v₁⁻¹v2v₃⁻¹)·Φ6(v0v₁⁻¹v₂⁻¹v3)

Φ4(x) =x²+ 1, Φ3(x) =x²+x+ 1, Φ6(x) =x²−x+ 1

Φ₄(1) = 2 Φ₃(1) = 3 Φ₆(1) = 1

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The example of G

2

χ_1,0, χ_1,6, χ_1,3⁰, χ_1,3⁰⁰, χ_2,1, χ_2,2.

s2 = 2·v₁²v₀⁻²·Φ3(v0v₁⁻¹v2v₃⁻¹)·Φ6(v0v₁⁻¹v₂⁻¹v3)

Φ4(x) =x²+ 1, Φ3(x) =x²+x+ 1, Φ6(x) =x²−x+ 1

Φ₄(1) = 2 Φ₃(1) = 3 Φ₆(1) = 1

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The example of G

2

χ_1,0, χ_1,6, χ_1,3⁰, χ_1,3⁰⁰, χ_2,1, χ_2,2.

s2 = 2·v₁²v₀⁻²·Φ3(v0v₁⁻¹v2v₃⁻¹)·Φ6(v0v₁⁻¹v₂⁻¹v3)

Φ4(x) =x²+ 1, Φ3(x) =x²+x+ 1, Φ6(x) =x²−x+ 1

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The 2-essential monomials for G₂ are:

M1 :=v0v₁⁻¹ and M2:=v2v₃⁻¹.

The 3-essential monomials for G2 are:

M₃ :=v₀v₁⁻¹v₂v₃⁻¹ and M₄:=v₀v₁⁻¹v₂⁻¹v₃.

Monomial B^M₂ (H) B^M₃ (H)

1 (χ_2,1, χ_2,2) -

M₁ (χ_1,0, χ_1,3⁰), (χ_2,1, χ_2,2), (χ_1,6, χ_1,3⁰⁰) - M₂ (χ_1,0, χ_1,3⁰⁰), (χ_2,1, χ_2,2), (χ_1,6, χ_1,3⁰) -

M₃ (χ_2,1, χ_2,2) (χ_1,0, χ_1,6, χ_2,2) M₄ (χ_2,1, χ_2,2) (χ_1,3⁰, χ_1,3⁰⁰, χ_2,1)

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M1 :=v0v₁⁻¹ and M2:=v2v₃⁻¹. The 3-essential monomials for G2 are:

M₃ :=v₀v₁⁻¹v₂v₃⁻¹ and M₄:=v₀v₁⁻¹v₂⁻¹v₃.

1 (χ_2,1, χ_2,2) -

M₁ (χ_1,0, χ_1,3⁰), (χ_2,1, χ_2,2), (χ_1,6, χ_1,3⁰⁰) - M₂ (χ_1,0, χ_1,3⁰⁰), (χ_2,1, χ_2,2), (χ_1,6, χ_1,3⁰) -

M₃ (χ_2,1, χ_2,2) (χ_1,0, χ_1,6, χ_2,2) M₄ (χ_2,1, χ_2,2) (χ_1,3⁰, χ_1,3⁰⁰, χ_2,1)

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M1 :=v0v₁⁻¹ and M2:=v2v₃⁻¹. The 3-essential monomials for G2 are:

M₃ :=v₀v₁⁻¹v₂v₃⁻¹ and M₄:=v₀v₁⁻¹v₂⁻¹v₃.

1 (χ_2,1, χ_2,2) -

M₁ (χ_1,0, χ_1,3⁰), (χ_2,1, χ_2,2), (χ_1,6, χ_1,3⁰⁰) - M₂ (χ_1,0, χ_1,3⁰⁰), (χ_2,1, χ_2,2), (χ_1,6, χ_1,3⁰) -

M₃ (χ_2,1, χ_2,2) (χ_1,0, χ_1,6, χ_2,2) M₄ (χ_2,1, χ_2,2) (χ_1,3⁰, χ_1,3⁰⁰, χ_2,1)

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Determination of the Rouquier blocks of a cyclotomic Hecke algebra

φ^s : v0 7→y v2 7→y v₁7→1 v₃ 7→1

The only essential monomial sent to 1 is M4. Thus the Rouquier blocks of H^s_φ are:

(χ1,0), (χ1,6), (χ1,3⁰, χ1,3⁰⁰, χ2,1, χ2,2).

Determination of the Rouquier blocks of the group algebra φ^W : v0 7→1 v2 7→1

v₁ 7→1 v₃ 7→1 All essential monomials are sent to 1. We have:

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ1,0, χ1,6, χ2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ2,1)

#1 Rouquier block (χ_1,0, χ_1,6, χ_1,3⁰, χ_1,3⁰⁰, χ_2,1, χ_2,2)

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Determination of the Rouquier blocks of a cyclotomic Hecke algebra φ^s : v₀ 7→y v₂ 7→y

v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ1,3⁰, χ1,3⁰⁰, χ2,1, χ2,2).

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ1,0, χ1,6, χ2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ2,1)

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v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ1,3⁰, χ1,3⁰⁰, χ2,1, χ2,2).

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ1,0, χ1,6, χ2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ2,1)

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v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ_1,3⁰, χ_1,3⁰⁰, χ2,1, χ2,2).

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ1,0, χ1,6, χ2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ2,1)

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v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ_1,3⁰, χ_1,3⁰⁰, χ2,1, χ2,2).

Determination of the Rouquier blocks of the group algebra

φ^W : v0 7→1 v2 7→1 v₁ 7→1 v₃ 7→1 All essential monomials are sent to 1. We have:

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ1,0, χ1,6, χ2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ2,1)

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v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ_1,3⁰, χ_1,3⁰⁰, χ2,1, χ2,2).

v₁ 7→1 v₃ 7→1

All essential monomials are sent to 1. We have:

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ1,0, χ1,6, χ2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ2,1)

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v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ_1,3⁰, χ_1,3⁰⁰, χ2,1, χ2,2).

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ1,0, χ1,6, χ2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ2,1)

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v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ_1,3⁰, χ_1,3⁰⁰, χ2,1, χ2,2).

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ1,0, χ1,6, χ2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ2,1)

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v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ_1,3⁰, χ_1,3⁰⁰, χ2,1, χ2,2).

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

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v₁7→1 v₃ 7→1

(χ1,0), (χ1,6), (χ_1,3⁰, χ_1,3⁰⁰, χ2,1, χ2,2).

#22-blocks (χ1,0, χ1,6, χ1,3⁰, χ1,3⁰⁰),(χ2,1, χ2,2)

#23-blocks (χ_1,0, χ_1,6, χ_2,2),(χ_1,3⁰, χ_1,3⁰⁰, χ_2,1)

#1Rouquier block (χ_1,0, χ_1,6, χ_1,3⁰, χ_1,3⁰⁰, χ_2,1, χ_2,2)