Families of characters of the imprimitive complex reflection groups

Families of characters of the imprimitive complex reflection groups

Maria Chlouveraki EPFL

Expansion of Combinatorial Representation Theory RIMS Workshop 2008

Complex reflection groups

Afinite reflection group onK is a finite subgroup ofGLK(V) (V a finite dimensionalK-vector space) generated by pseudo-reflections,i.e.,linear maps whose vector space of fixed points is a hyperplane.

A finite reflection group onQis called aWeyl group.

A finite reflection group onRis called a(finite) Coxeter group. A finite reflection group onCis called acomplex reflection group.

Complex reflection groups

Afinite reflection group onK is a finite subgroup ofGLK(V) (V a finite dimensionalK-vector space) generated by pseudo-reflections,i.e.,linear maps whose vector space of fixed points is a hyperplane.

A finite reflection group onQis called aWeyl group.

A finite reflection group onRis called a(finite) Coxeter group. A finite reflection group onCis called acomplex reflection group.

Complex reflection groups

Afinite reflection group onK is a finite subgroup ofGLK(V) (V a finite dimensionalK-vector space) generated by pseudo-reflections,i.e.,linear maps whose vector space of fixed points is a hyperplane.

A finite reflection group onQis called aWeyl group.

A finite reflection group onRis called a(finite) Coxeter group. A finite reflection group onCis called acomplex reflection group.

Complex reflection groups

Afinite reflection group onK is a finite subgroup ofGLK(V) (V a finite dimensionalK-vector space) generated by pseudo-reflections,i.e.,linear maps whose vector space of fixed points is a hyperplane.

A finite reflection group onQis called aWeyl group.

A finite reflection group onRis called a(finite) Coxeter group.

A finite reflection group onCis called acomplex reflection group.

Complex reflection groups

Afinite reflection group onK is a finite subgroup ofGLK(V) (V a finite dimensionalK-vector space) generated by pseudo-reflections,i.e.,linear maps whose vector space of fixed points is a hyperplane.

A finite reflection group onQis called aWeyl group.

A finite reflection group onRis called a(finite) Coxeter group.

A finite reflection group onCis called acomplex reflection group.

The complex reflection groups were classified by Shephard and Todd in 1954. If W is an (irreducible) complex reflection group, then

either there exist positive integers d,e,r such that W is isomorphic to G(de,e,r), whereG(de,e,r) is the group of allr×r monomial matrices with non-zero entries inµde and product of the non-zero entries inµd, orW is isomorphic to an exceptional groupGn (n= 4, . . . ,37).

The complex reflection groups were classified by Shephard and Todd in 1954. If W is an (irreducible) complex reflection group, then

either there exist positive integersd,e,r such that W is isomorphic to G(de,e,r), whereG(de,e,r) is the group of allr×r monomial matrices with non-zero entries inµde and product of the non-zero entries inµd,

orW is isomorphic to an exceptional groupGn (n= 4, . . . ,37).

The complex reflection groups were classified by Shephard and Todd in 1954. If W is an (irreducible) complex reflection group, then

either there exist positive integersd,e,r such that W is isomorphic to G(de,e,r), whereG(de,e,r) is the group of allr×r monomial matrices with non-zero entries inµde and product of the non-zero entries inµd, orW is isomorphic to an exceptional groupGn (n= 4, . . . ,37).

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation: G2=<s,t|ststst=tststs,s²=t²= 1>,

G₄=<s,t|sts=tst,s³=t³= 1>, and afield of realizationK:

KG₂ =Q, KG₄=Q(ζ3).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where s runs over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0>, H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0,

(T −u0)(T −u1)(T −u2) = 0> .

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation:

G2=<s,t|ststst=tststs,s²=t²= 1>, G₄=<s,t|sts=tst,s³=t³= 1>, and afield of realizationK:

KG₂ =Q, KG₄=Q(ζ3).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where s runs over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0>, H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0,

(T −u0)(T −u1)(T −u2) = 0> .

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation: G2=<s,t|ststst=tststs,s²=t²= 1>,

G₄=<s,t|sts=tst,s³=t³= 1>,

and afield of realizationK:

KG₂ =Q, KG₄=Q(ζ3).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where s runs over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0>, H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0,

(T −u0)(T −u1)(T −u2) = 0> .

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation: G2=<s,t|ststst=tststs,s²=t²= 1>,

G₄=<s,t|sts=tst,s³=t³= 1>, and afield of realizationK:

KG₂ =Q, KG₄=Q(ζ3).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where s runs over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0>, H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0,

(T −u0)(T −u1)(T −u2) = 0> .

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation: G2=<s,t|ststst=tststs,s²=t²= 1>,

G₄=<s,t|sts=tst,s³=t³= 1>, and afield of realizationK:

K_G2 =Q, K_G4=Q(ζ₃).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where s runs over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0>, H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0,

(T −u0)(T −u1)(T −u2) = 0> .

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation: G2=<s,t|ststst=tststs,s²=t²= 1>,

G₄=<s,t|sts=tst,s³=t³= 1>, and afield of realizationK:

K_G2 =Q, K_G4=Q(ζ₃).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where sruns over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0>, H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0,

(T −u0)(T −u1)(T −u2) = 0> .

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation: G2=<s,t|ststst=tststs,s²=t²= 1>,

G₄=<s,t|sts=tst,s³=t³= 1>, and afield of realizationK:

K_G2 =Q, K_G4=Q(ζ₃).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where sruns over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0>, H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0,

(T −u0)(T −u1)(T −u2) = 0> .

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation: G2=<s,t|ststst=tststs,s²=t²= 1>,

G₄=<s,t|sts=tst,s³=t³= 1>, and afield of realizationK:

K_G2 =Q, K_G4=Q(ζ₃).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where sruns over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w )(T −w ) = 0>,

H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0, (T −u0)(T −u1)(T −u2) = 0> .

Generic Hecke algebras

Every complex reflection groupW has anCoxeter-like presentation: G2=<s,t|ststst=tststs,s²=t²= 1>,

G₄=<s,t|sts=tst,s³=t³= 1>, and afield of realizationK:

K_G2 =Q, K_G4=Q(ζ₃).

We choose a set of indeterminatesu= (us,j)_{s,0≤j≤o(s)−1}, where sruns over the set of generators ofW ando(s) denotes the order ofs (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0>, H(G4) =<S,T|STS=TST, (S−u0)(S−u1)(S−u2) = 0,

Remark: The specializationus,j 7→ζ_o(s)^j sendsH(W) toZKW.

Theorem (Malle)

Let v= (vs,j)s,j be a set of indeterminates such that, for alls,j, we have v_s^|µ(K)|_,j :=ζ_o(s^−j₎us,j,

where µ(K) is the group of all the roots of unity inK. Then theK(v)-algebra K(v)H(W) is split semisimple.

By “Tits’ deformation theorem”, the specialization v_s,j 7→1 induces a bijection Irr(K(v)H(W)) ↔ Irr(W)

χ_v 7→ χ

Remark: The specializationus,j 7→ζ_o(s)^j sendsH(W) toZKW.

Theorem (Malle)

Letv= (vs,j)s,j be a set of indeterminates such that, for alls,j, we have v_s,j^|µ(K)|:=ζ_o(s^−j₎us,j,

where µ(K) is the group of all the roots of unity inK. Then theK(v)-algebra K(v)H(W) is split semisimple.

By “Tits’ deformation theorem”, the specialization v_s,j 7→1 induces a bijection Irr(K(v)H(W)) ↔ Irr(W)

χ_v 7→ χ

Remark: The specializationus,j 7→ζ_o(s)^j sendsH(W) toZKW.

Theorem (Malle)

Letv= (vs,j)s,j be a set of indeterminates such that, for alls,j, we have v_s,j^|µ(K)|:=ζ_o(s^−j₎us,j,

where µ(K) is the group of all the roots of unity inK. Then theK(v)-algebra K(v)H(W) is split semisimple.

By “Tits’ deformation theorem”, the specialization v_s,j 7→1 induces a bijection Irr(K(v)H(W)) ↔ Irr(W)

χ 7→ χ

Generic Schur elements

The generic Hecke algebra is endowed with acanonical symmetrizing formt. We have that

t = X

χ∈Irr(W)

1 sχ

χv,

where sχ is theSchur elementassociated toχv ∈Irr(K(v)H(W)).

Theorem (C.)

Let χ∈Irr(W). The Schur elements_χ is an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form: Ψ(M)

where

Ψis aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0,i.e.,ifM=Q

s,jv_s,j^as,j, then gcd(as,j) = 1 andP

s,jas,j = 0.

Generic Schur elements

The generic Hecke algebra is endowed with acanonical symmetrizing formt. We have that

t = X

χ∈Irr(W)

1 sχ

χv,

where sχ is theSchur elementassociated toχv ∈Irr(K(v)H(W)).

Theorem (C.)

Letχ∈Irr(W). The Schur elements_χ is an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form: Ψ(M)

where

Ψis aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0,i.e.,ifM=Q

s,jv_s,j^as,j, then gcd(as,j) = 1 andP

s,jas,j = 0.

Generic Schur elements

The generic Hecke algebra is endowed with acanonical symmetrizing formt. We have that

t = X

χ∈Irr(W)

1 sχ

χv,

where sχ is theSchur elementassociated toχv ∈Irr(K(v)H(W)).

Theorem (C.)

Letχ∈Irr(W). The Schur elements_χ is an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form: Ψ(M)

where

Ψis aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0,i.e.,ifM=Q

s,jv_s,j^as,j, then gcd(as,j) = 1 andP

s,jas,j = 0.

Generic Schur elements

The generic Hecke algebra is endowed with acanonical symmetrizing formt. We have that

t = X

χ∈Irr(W)

1 sχ

χv,

where sχ is theSchur elementassociated toχv ∈Irr(K(v)H(W)).

Theorem (C.)

Letχ∈Irr(W). The Schur elements_χ is an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form: Ψ(M)

where

Ψis aK-cyclotomic polynomial in one variable,

M is a primitive monomial of degree 0,i.e.,ifM=Q

s,jv_s,j^as,j, then gcd(as,j) = 1 andP

s,jas,j = 0.

Generic Schur elements

The generic Hecke algebra is endowed with acanonical symmetrizing formt. We have that

t = X

χ∈Irr(W)

1 sχ

χv,

where sχ is theSchur elementassociated toχv ∈Irr(K(v)H(W)).

Theorem (C.)

Letχ∈Irr(W). The Schur elements_χ is an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form: Ψ(M)

where

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

i.e.,ifM=Q

s,jv_s,j^as,j, then gcd(as,j) = 1 andP

s,jas,j = 0.

Generic Schur elements

The generic Hecke algebra is endowed with acanonical symmetrizing formt. We have that

t = X

χ∈Irr(W)

1 sχ

χv,

where sχ is theSchur elementassociated toχv ∈Irr(K(v)H(W)).

Theorem (C.)

Letχ∈Irr(W). The Schur elements_χ is an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form: Ψ(M)

where

Ψis aK-cyclotomic polynomial in one variable,

Q a

Schur elements of G

: X

:= u

, X

:= −u

, Y

:= w

, Y

:= −w

.

s1= Φ4(X0X₁⁻¹)·Φ4(Y0Y₁⁻¹)·Φ3(X0Y0X₁⁻¹Y₁⁻¹)·Φ6(X0Y0X₁⁻¹Y₁⁻¹)

s2= 2·X₁²X₀⁻²·Φ3(X0Y0X₁⁻¹Y₁⁻¹)·Φ6(X0Y1X₁⁻¹Y₀⁻¹)

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

Cyclotomic Hecke algebras

Lety be an indeterminate. Acyclotomic specializationofHis a ZK-algebra morphismφ:ZK[v,v⁻¹]→ZK[y,y⁻¹] of the form:

φ:v_s,j7→y^ns,j wheren_s,j∈Zfor allsandj.

The correspondingcyclotomic Hecke algebraHφis the ZK[y,y⁻¹]-algebra obtained as the specialization ofH(W) via the morphismφ.

Ifq:=y^|µ(K)|, then the morphismφcan be also described as follows: φ:us,j 7→ζ_o(s)^j q^ns,j.

Proposition (C.)

The algebraK(y)Hφ is split semisimple.

Cyclotomic Hecke algebras

Lety be an indeterminate. Acyclotomic specializationofHis a ZK-algebra morphismφ:ZK[v,v⁻¹]→ZK[y,y⁻¹] of the form:

φ:v_s,j7→y^ns,j wheren_s,j∈Zfor allsandj.

The correspondingcyclotomic Hecke algebraHφis theZK[y,y⁻¹]-algebra obtained as the specialization ofH(W) via the morphismφ.

Ifq:=y^|µ(K)|, then the morphismφcan be also described as follows: φ:us,j 7→ζ_o(s)^j q^ns,j.

Proposition (C.)

The algebraK(y)Hφ is split semisimple.

Cyclotomic Hecke algebras

Lety be an indeterminate. Acyclotomic specializationofHis a ZK-algebra morphismφ:ZK[v,v⁻¹]→ZK[y,y⁻¹] of the form:

φ:v_s,j7→y^ns,j wheren_s,j∈Zfor allsandj.

The correspondingcyclotomic Hecke algebraHφis theZK[y,y⁻¹]-algebra obtained as the specialization ofH(W) via the morphismφ.

Ifq:=y^|µ(K)|, then the morphismφcan be also described as follows:

φ:us,j 7→ζ_o(s)^j q^ns,j.

Proposition (C.)

The algebraK(y)Hφ is split semisimple.

Cyclotomic Hecke algebras

Lety be an indeterminate. Acyclotomic specializationofHis a ZK-algebra morphismφ:ZK[v,v⁻¹]→ZK[y,y⁻¹] of the form:

φ:v_s,j7→y^ns,j wheren_s,j∈Zfor allsandj.

The correspondingcyclotomic Hecke algebraHφis theZK[y,y⁻¹]-algebra obtained as the specialization ofH(W) via the morphismφ.

Ifq:=y^|µ(K)|, then the morphismφcan be also described as follows:

φ:us,j 7→ζ_o(s)^j q^ns,j.

Proposition (C.)

The algebraK(y)Hφ is split semisimple.

By “Tits’ deformation theorem”, we obtain that the specialization vs,j 7→1 induces the following bijections :

Irr(K(v)H) ↔ Irr(K(y)Hφ) ↔ Irr(W)

χv 7→ χφ 7→ χ

Proposition

The Schur element sχ_φ(y)associated to the irreducible characterχφ ofK(y)Hφ

is a Laurent polynomial iny of the form sχ_φ(y) =ψχ_φy^aχφ Y

Φ∈C_K

Φ(y)^nχφ,Φ,

where ψχ_φ ∈ZK,aχ_φ ∈Z, nχ_φ,Φ∈NandCK is a set ofK-cyclotomic polynomials.

By “Tits’ deformation theorem”, we obtain that the specialization vs,j 7→1 induces the following bijections :

Irr(K(v)H) ↔ Irr(K(y)Hφ) ↔ Irr(W)

χv 7→ χφ 7→ χ

Proposition

The Schur element sχ_φ(y)associated to the irreducible characterχφ ofK(y)Hφ

is a Laurent polynomial iny of the form sχ_φ(y) =ψχ_φy^aχφ Y

Φ∈C_K

Φ(y)^nχφ,Φ,

where ψχ_φ ∈ZK,aχ_φ ∈Z, nχ_φ,Φ∈NandCK is a set ofK-cyclotomic polynomials.

Rouquier blocks of H

TheRouquier blocksof the cyclotomic Hecke algebraH_φ are the blocks of the algebra R_K(y)H_φ, where

RK(y) :=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

i.e.,the partitionRB(Hφ) ofIrr(W) minimal for the property: For allB∈ RB(Hφ)andh∈ Hφ,X

χ∈B

χφ(h)

s_χφ ∈ RK(y).

W Weyl group : Rouquier blocks≡ families of characters W c.r.g. (non-Weyl) : Rouquier blocks≡“families of characters”

Rouquier blocks of H

TheRouquier blocksof the cyclotomic Hecke algebraH_φ are the blocks of the algebra R_K(y)H_φ, where

RK(y) :=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

i.e.,the partitionRB(Hφ) ofIrr(W) minimal for the property:

For allB∈ RB(Hφ)andh∈ Hφ,X

χ∈B

χφ(h)

s_χφ ∈ RK(y).

W Weyl group : Rouquier blocks≡ families of characters W c.r.g. (non-Weyl) : Rouquier blocks≡“families of characters”

Rouquier blocks of H

TheRouquier blocksof the cyclotomic Hecke algebraH_φ are the blocks of the algebra R_K(y)H_φ, where

RK(y) :=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

i.e.,the partitionRB(Hφ) ofIrr(W) minimal for the property:

For allB∈ RB(Hφ)andh∈ Hφ,X

χ∈B

χφ(h)

s_χφ ∈ RK(y).

W Weyl group : Rouquier blocks≡ families of characters

W c.r.g. (non-Weyl) : Rouquier blocks≡“families of characters”

Rouquier blocks of H

TheRouquier blocksof the cyclotomic Hecke algebraH_φ are the blocks of the algebra R_K(y)H_φ, where

RK(y) :=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

i.e.,the partitionRB(Hφ) ofIrr(W) minimal for the property:

For allB∈ RB(Hφ)andh∈ Hφ,X

χ∈B

χφ(h)

s_χφ ∈ RK(y).

W Weyl group : Rouquier blocks≡ families of characters W c.r.g. (non-Weyl) : Rouquier blocks≡“families of characters”

Essential monomials and essential hyperplanes

Letp be a prime ideal ofZK.

A primitive monomialM in ZK[v,v⁻¹] is calledp-essential forW if there exist an irreducible character χofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividessχ(v)

2 Ψ(1)∈p.

A primitive monomialM is calledessential forW if it is p-essential for some prime ideal pofZK.

Essential monomials and essential hyperplanes

Letp be a prime ideal ofZK.

A primitive monomialM inZK[v,v⁻¹] is calledp-essential forW if there exist an irreducible character χofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividessχ(v)

2 Ψ(1)∈p.

A primitive monomialM is calledessential forW if it is p-essential for some prime ideal pofZK.

Essential monomials and essential hyperplanes

Letp be a prime ideal ofZK.

A primitive monomialM inZK[v,v⁻¹] is calledp-essential forW if there exist an irreducible character χofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividess_χ(v)

2 Ψ(1)∈p.

A primitive monomialM is calledessential forW if it is p-essential for some prime ideal pofZK.

Essential monomials and essential hyperplanes

Letp be a prime ideal ofZK.

A primitive monomialM inZK[v,v⁻¹] is calledp-essential forW if there exist an irreducible character χofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividess_χ(v)

2 Ψ(1)∈p.

A primitive monomialM is calledessential forW if it is p-essential for some prime ideal pofZK.

Essential monomials and essential hyperplanes

Letp be a prime ideal ofZK.

A primitive monomialM inZK[v,v⁻¹] is calledp-essential forW if there exist an irreducible character χofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividess_χ(v)

2 Ψ(1)∈p.

A primitive monomialM is calledessential forW if it is p-essential for some prime ideal pofZK.

Schur elements of G

: 2-essential in purple, 3-essential in green.

s1= Φ4(X0X₁⁻¹)·Φ4(Y0Y₁⁻¹)·Φ3(X0Y0X₁⁻¹Y₁⁻¹)·Φ6(X0Y0X₁⁻¹Y₁⁻¹)

s₂= 2·X₁²X₀⁻²·Φ₃(X₀Y₀X₁⁻¹Y₁⁻¹)·Φ₆(X₀Y₁X₁⁻¹Y₀⁻¹)

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

Φ₄(1) = 2, Φ₃(1) = 3, Φ₆(1) = 1.

Letφ:v_s,j 7→y^ns,j be a cyclotomic specialization and letM=Q

s,jv_s,j^as,j be an essential monomial forW.

We have

φ(M) = 1⇔X

s,j

as,jns,j = 0.

The hyperplaneP

s,jas,jts,j = 0 is called anessential hyperplane forW. If the integersn_s,j belong to no essential hyperplane, then the Rouquier blocks ofHφ are calledRouquier blocks associated with no essential hyperplane.

If the integersn_s,j belong to exactly one essential hyperplaneH, then the Rouquier blocks of Hφ are calledRouquier blocks associated with the essential hyperplane H.

Theorem (C.)

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization. The Rouquier blocks ofHφ is a partition generated by the Rouquier blocks associated with the essential hyperplanes that the ns,j belong to.

Letφ:v_s,j 7→y^ns,j be a cyclotomic specialization and letM=Q

s,jv_s,j^as,j be an essential monomial forW. We have

φ(M) = 1⇔X

s,j

as,jns,j = 0.

The hyperplaneP

s,jas,jts,j = 0 is called anessential hyperplane forW. If the integersn_s,j belong to no essential hyperplane, then the Rouquier blocks ofHφ are calledRouquier blocks associated with no essential hyperplane.

If the integersn_s,j belong to exactly one essential hyperplaneH, then the Rouquier blocks of Hφ are calledRouquier blocks associated with the essential hyperplane H.

Theorem (C.)

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization. The Rouquier blocks ofHφ is a partition generated by the Rouquier blocks associated with the essential hyperplanes that the ns,j belong to.

Letφ:v_s,j 7→y^ns,j be a cyclotomic specialization and letM=Q

s,jv_s,j^as,j be an essential monomial forW. We have

φ(M) = 1⇔X

s,j

as,jns,j = 0.

The hyperplaneP

s,jas,jts,j = 0 is called anessential hyperplane forW.

If the integersn_s,j belong to no essential hyperplane, then the Rouquier blocks ofHφ are calledRouquier blocks associated with no essential hyperplane.

If the integersn_s,j belong to exactly one essential hyperplaneH, then the Rouquier blocks of Hφ are calledRouquier blocks associated with the essential hyperplane H.

Theorem (C.)

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization. The Rouquier blocks ofHφ is a partition generated by the Rouquier blocks associated with the essential hyperplanes that the ns,j belong to.

Letφ:v_s,j 7→y^ns,j be a cyclotomic specialization and letM=Q

s,jv_s,j^as,j be an essential monomial forW. We have

φ(M) = 1⇔X

s,j

as,jns,j = 0.

The hyperplaneP

s,jas,jts,j = 0 is called anessential hyperplane forW. If the integersns,j belong to no essential hyperplane, then the Rouquier blocks ofHφare calledRouquier blocks associated with no essential hyperplane.

If the integersn_s,j belong to exactly one essential hyperplaneH, then the Rouquier blocks of H_φ are calledRouquier blocks associated with the essential hyperplane H.

Theorem (C.)

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization. The Rouquier blocks ofHφ is a partition generated by the Rouquier blocks associated with the essential hyperplanes that the ns,j belong to.

Letφ:v_s,j 7→y^ns,j be a cyclotomic specialization and letM=Q

s,jv_s,j^as,j be an essential monomial forW. We have

φ(M) = 1⇔X

s,j

as,jns,j = 0.

The hyperplaneP

s,jas,jts,j = 0 is called anessential hyperplane forW. If the integersns,j belong to no essential hyperplane, then the Rouquier blocks ofHφare calledRouquier blocks associated with no essential hyperplane.

If the integersn_s,j belong to exactly one essential hyperplaneH, then the Rouquier blocks of H_φ are calledRouquier blocks associated with the essential hyperplane H.

Theorem (C.)

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization. The Rouquier blocks ofHφ is a partition generated by the Rouquier blocks associated with the essential hyperplanes that the ns,j belong to.

Letφ:v_s,j 7→y^ns,j be a cyclotomic specialization and letM=Q

s,jv_s,j^as,j be an essential monomial forW. We have

φ(M) = 1⇔X

s,j

as,jns,j = 0.

The hyperplaneP

s,jas,jts,j = 0 is called anessential hyperplane forW. If the integersns,j belong to no essential hyperplane, then the Rouquier blocks ofHφare calledRouquier blocks associated with no essential hyperplane.

If the integersn_s,j belong to exactly one essential hyperplaneH, then the Rouquier blocks of H_φ are calledRouquier blocks associated with the essential hyperplane H.

Theorem (C.)

Letφ:vs,j 7→y^ns,j be a cyclotomic specialization. The Rouquier blocks ofHφ is a partition generated by the Rouquier blocks associated with the essential

Combinatorics

Letλ= (λ1, λ2, . . . , λh) be a partition,i.e.,a finite decreasing sequence of positive integers:

λ1≥λ2≥. . .≥λh≥1.

The integer|λ|:=λ₁+λ₂+. . .+λ_his called the size ofλ. We also say thatλis a partition of |λ|. The integerhis calledthe height ofλand we seth_λ:=h. To each partition λwe associate itsβ-number,β_λ= (β₁, β₂, . . . , β_h), defined by

β1:=h+λ1−1, β2:=h+λ2−2, . . . , βh:=h+λh−h. Example: Ifλ= (4,2,2,1), thenβλ= (7,4,3,1).

Let m∈N. Them-shiftedβ-numberofλis the sequence of numbers defined by β_λ[m] = (β₁+m, β₂+m, . . . , β_h+m,m−1,m−2, . . . ,1,0).

Example: Ifλ= (4,2,2,1), thenβλ[3] = (10,7,6,4,2,1,0).

Combinatorics

Letλ= (λ1, λ2, . . . , λh) be a partition,i.e.,a finite decreasing sequence of positive integers:

λ1≥λ2≥. . .≥λh≥1.

The integer|λ|:=λ₁+λ₂+. . .+λ_his calledthe size ofλ. We also say thatλis a partition of |λ|.

The integerhis calledthe height ofλand we seth_λ:=h. To each partition λwe associate itsβ-number,β_λ= (β₁, β₂, . . . , β_h), defined by

β1:=h+λ1−1, β2:=h+λ2−2, . . . , βh:=h+λh−h. Example: Ifλ= (4,2,2,1), thenβλ= (7,4,3,1).

Let m∈N. Them-shiftedβ-numberofλis the sequence of numbers defined by β_λ[m] = (β₁+m, β₂+m, . . . , β_h+m,m−1,m−2, . . . ,1,0).

Example: Ifλ= (4,2,2,1), thenβλ[3] = (10,7,6,4,2,1,0).

Combinatorics

Letλ= (λ1, λ2, . . . , λh) be a partition,i.e.,a finite decreasing sequence of positive integers:

λ1≥λ2≥. . .≥λh≥1.

The integer|λ|:=λ₁+λ₂+. . .+λ_his calledthe size ofλ. We also say thatλis a partition of |λ|. The integerhis calledthe height ofλand we seth_λ:=h.

To each partition λwe associate itsβ-number,β_λ= (β₁, β₂, . . . , β_h), defined by β1:=h+λ1−1, β2:=h+λ2−2, . . . , βh:=h+λh−h.

Example: Ifλ= (4,2,2,1), thenβλ= (7,4,3,1).

Let m∈N. Them-shiftedβ-numberofλis the sequence of numbers defined by β_λ[m] = (β₁+m, β₂+m, . . . , β_h+m,m−1,m−2, . . . ,1,0).

Example: Ifλ= (4,2,2,1), thenβλ[3] = (10,7,6,4,2,1,0).

Combinatorics

Letλ= (λ1, λ2, . . . , λh) be a partition,i.e.,a finite decreasing sequence of positive integers:

λ1≥λ2≥. . .≥λh≥1.

The integer|λ|:=λ₁+λ₂+. . .+λ_his calledthe size ofλ. We also say thatλis a partition of |λ|. The integerhis calledthe height ofλand we seth_λ:=h.

To each partition λwe associate itsβ-number,β_λ= (β₁, β₂, . . . , β_h), defined by β1:=h+λ1−1, β2:=h+λ2−2, . . . , βh:=h+λh−h.

Example: Ifλ= (4,2,2,1), thenβλ= (7,4,3,1).

Let m∈N. Them-shiftedβ-numberofλis the sequence of numbers defined by β_λ[m] = (β₁+m, β₂+m, . . . , β_h+m,m−1,m−2, . . . ,1,0).

Example: Ifλ= (4,2,2,1), thenβλ[3] = (10,7,6,4,2,1,0).

Combinatorics

Letλ= (λ1, λ2, . . . , λh) be a partition,i.e.,a finite decreasing sequence of positive integers:

λ1≥λ2≥. . .≥λh≥1.

The integer|λ|:=λ₁+λ₂+. . .+λ_his calledthe size ofλ. We also say thatλis a partition of |λ|. The integerhis calledthe height ofλand we seth_λ:=h.

To each partition λwe associate itsβ-number,β_λ= (β₁, β₂, . . . , β_h), defined by β1:=h+λ1−1, β2:=h+λ2−2, . . . , βh:=h+λh−h.

Example: Ifλ= (4,2,2,1), thenβλ= (7,4,3,1).

Let m∈N. Them-shiftedβ-numberofλis the sequence of numbers defined by β_λ[m] = (β₁+m, β₂+m, . . . , β_h+m,m−1,m−2, . . . ,1,0).

Example: Ifλ= (4,2,2,1), thenβλ[3] = (10,7,6,4,2,1,0).

Combinatorics

Letλ= (λ1, λ2, . . . , λh) be a partition,i.e.,a finite decreasing sequence of positive integers:

λ1≥λ2≥. . .≥λh≥1.

The integer|λ|:=λ₁+λ₂+. . .+λ_his calledthe size ofλ. We also say thatλis a partition of |λ|. The integerhis calledthe height ofλand we seth_λ:=h.

To each partition λwe associate itsβ-number,β_λ= (β₁, β₂, . . . , β_h), defined by β1:=h+λ1−1, β2:=h+λ2−2, . . . , βh:=h+λh−h.

Example: Ifλ= (4,2,2,1), thenβλ= (7,4,3,1).

Letm∈N. Them-shiftedβ-numberofλis the sequence of numbers defined by β_λ[m] = (β₁+m, β₂+m, . . . , β_h+m,m−1,m−2, . . . ,1,0).

Example: Ifλ= (4,2,2,1), thenβλ[3] = (10,7,6,4,2,1,0).

Combinatorics

Letλ= (λ1, λ2, . . . , λh) be a partition,i.e.,a finite decreasing sequence of positive integers:

λ1≥λ2≥. . .≥λh≥1.

The integer|λ|:=λ₁+λ₂+. . .+λ_his calledthe size ofλ. We also say thatλis a partition of |λ|. The integerhis calledthe height ofλand we seth_λ:=h.

To each partition λwe associate itsβ-number,β_λ= (β₁, β₂, . . . , β_h), defined by β1:=h+λ1−1, β2:=h+λ2−2, . . . , βh:=h+λh−h.

Example: Ifλ= (4,2,2,1), thenβλ= (7,4,3,1).

Letm∈N. Them-shiftedβ-numberofλis the sequence of numbers defined by

Letd be a positive integer. A family ofd partitionsλ= (λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(d−1)) is called a d-partition.

We set

h^(a):=h_λ(a), β^(a):=β_λ(a)

and we have

λ^(a) = (λ^(a)₁ , λ^(a)₂ , . . . , λ^(a)_h(a)). The integer

|λ|:=

d−1

X

a=0

|λ^(a)|

is calledthe size ofλ. We also say thatλis ad-partition of|λ|.

From now on, we suppose that we have a given “weight system”, i.e.,a family of integers

m:= (m⁽⁰⁾,m⁽¹⁾, . . . ,m^(d−1)).

Letd be a positive integer. A family ofd partitionsλ= (λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(d−1)) is called a d-partition. We set

h^(a):=h_λ(a), β^(a):=β_λ(a)

and we have

λ^(a) = (λ^(a)₁ , λ^(a)₂ , . . . , λ^(a)_h(a)).

The integer

|λ|:=

d−1

X

a=0

|λ^(a)|

is calledthe size ofλ. We also say thatλis ad-partition of|λ|.

From now on, we suppose that we have a given “weight system”, i.e.,a family of integers

m:= (m⁽⁰⁾,m⁽¹⁾, . . . ,m^(d−1)).

Letd be a positive integer. A family ofd partitionsλ= (λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(d−1)) is called a d-partition. We set

h^(a):=h_λ(a), β^(a):=β_λ(a)

and we have

λ^(a) = (λ^(a)₁ , λ^(a)₂ , . . . , λ^(a)_h(a)).

The integer

|λ|:=

d−1

X

a=0

|λ^(a)|

is calledthe size ofλ. We also say thatλis ad-partition of|λ|.

From now on, we suppose that we have a given “weight system”, i.e.,a family of integers

m:= (m⁽⁰⁾,m⁽¹⁾, . . . ,m^(d−1)).

Letd be a positive integer. A family ofd partitionsλ= (λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(d−1)) is called a d-partition. We set

h^(a):=h_λ(a), β^(a):=β_λ(a)

and we have

λ^(a) = (λ^(a)₁ , λ^(a)₂ , . . . , λ^(a)_h(a)).

The integer

|λ|:=

d−1

X

a=0

|λ^(a)|

is calledthe size ofλ. We also say thatλis ad-partition of|λ|.

From now on, we suppose that we have a given “weight system”, i.e.,a family of integers

We define them-charged height ofλto be the integer hcλ:=max{hc^(a)|(0≤a≤d−1)}, where

hc⁽⁰⁾:=h⁽⁰⁾−m⁽⁰⁾,hc⁽¹⁾:=h⁽¹⁾−m⁽¹⁾, . . . ,hc^(d−1):=h^(d−1)−m^(d−1).

Definition (m-charged standard symbol and content)

Them-charged standard symbolofλis the family of numbers defined by Bc_λ= (Bc_λ⁽⁰⁾,Bc_λ⁽¹⁾, . . . ,Bc_λ^(d−1)),

where, for all a(0≤a≤d−1), we have

Bc_λ^(a):=β^(a)[hcλ−hc^(a)].

Them-charged contentofλis the multiset

Contc_λ=Bc_λ⁽⁰⁾∪Bc_λ⁽¹⁾∪. . .∪Bc_λ^(d−1).

We define them-charged height ofλto be the integer hcλ:=max{hc^(a)|(0≤a≤d−1)}, where

hc⁽⁰⁾:=h⁽⁰⁾−m⁽⁰⁾,hc⁽¹⁾:=h⁽¹⁾−m⁽¹⁾, . . . ,hc^(d−1):=h^(d−1)−m^(d−1).

Definition (m-charged standard symbol and content)

Them-charged standard symbolofλis the family of numbers defined by Bcλ= (Bc_λ⁽⁰⁾,Bc_λ⁽¹⁾, . . . ,Bc_λ^(d−1)),

where, for all a(0≤a≤d−1), we have

Bc_λ^(a) :=β^(a)[hc_λ−hc^(a)].

Them-charged contentofλis the multiset

Contc_λ=Bc_λ⁽⁰⁾∪Bc_λ⁽¹⁾∪. . .∪Bc_λ^(d−1).

We define them-charged height ofλto be the integer hcλ:=max{hc^(a)|(0≤a≤d−1)}, where

hc⁽⁰⁾:=h⁽⁰⁾−m⁽⁰⁾,hc⁽¹⁾:=h⁽¹⁾−m⁽¹⁾, . . . ,hc^(d−1):=h^(d−1)−m^(d−1).

Definition (m-charged standard symbol and content)

Them-charged standard symbolofλis the family of numbers defined by Bcλ= (Bc_λ⁽⁰⁾,Bc_λ⁽¹⁾, . . . ,Bc_λ^(d−1)),

where, for all a(0≤a≤d−1), we have

Bc_λ^(a) :=β^(a)[hc_λ−hc^(a)].

Them-charged contentofλis the multiset

Example: Let us taked= 2, λ= ((3),(2,1)) andm= (2,−1). Then β⁽⁰⁾= (3),

β⁽¹⁾= (3,1), hc⁽⁰⁾= 1−2 =−1, hc⁽¹⁾= 2−(−1) = 3

=hcλ

.

Consequently,

Bcλ=

7 3 2 1 0

3 1

.

We haveContcλ={0,1,1,2,3,3,7}.

Example: Let us taked= 2, λ= ((3),(2,1)) andm= (2,−1). Then β⁽⁰⁾= (3),

β⁽¹⁾= (3,1), hc⁽⁰⁾= 1−2 =−1,

hc⁽¹⁾= 2−(−1) = 3 =hcλ.

Consequently,

Bcλ=

7 3 2 1 0

3 1

.

We haveContcλ={0,1,1,2,3,3,7}.

Example: Let us taked= 2, λ= ((3),(2,1)) andm= (2,−1). Then β⁽⁰⁾= (3),

β⁽¹⁾= (3,1), hc⁽⁰⁾= 1−2 =−1,

hc⁽¹⁾= 2−(−1) = 3 =hcλ.

Consequently,

Bcλ=

7 3 2 1 0

3 1

.

We haveContcλ={0,1,1,2,3,3,7}.

Example: Let us taked= 2, λ= ((3),(2,1)) andm= (2,−1). Then β⁽⁰⁾= (3),

β⁽¹⁾= (3,1), hc⁽⁰⁾= 1−2 =−1,

hc⁽¹⁾= 2−(−1) = 3 =hcλ.

Consequently,

Bcλ=

7 3 2 1 0

3 1

.

We haveContcλ={0,1,1,2,3,3,7}.

The group G (d , 1, r )

The groupG(d,1,r) is the group of allr×r monomial matrices whose non-zero entries lie in µ_d.

G(d,1,r)'µdoSr.

The irreducible characters of G(d,1,r) are indexed by thed-partitions ofr. The field of definition ofG(d,1,r) isQ(ζd).

G(1,1,r)'A_r−1forr≥2,

G(2,1,r)'Br forr ≥2 (G(2,1,1)'C2).

The group G (d , 1, r )

The groupG(d,1,r) is the group of allr×r monomial matrices whose non-zero entries lie in µ_d.

G(d,1,r)'µdoSr.

The irreducible characters of G(d,1,r) are indexed by thed-partitions ofr. The field of definition ofG(d,1,r) isQ(ζd).

G(1,1,r)'A_r−1forr≥2,

G(2,1,r)'Br forr ≥2 (G(2,1,1)'C2).

The group G (d , 1, r )

The groupG(d,1,r) is the group of allr×r monomial matrices whose non-zero entries lie in µ_d.

G(d,1,r)'µdoSr.

The irreducible characters of G(d,1,r) are indexed by thed-partitions ofr.

The field of definition ofG(d,1,r) isQ(ζd).

G(1,1,r)'A_r−1forr≥2,

G(2,1,r)'Br forr ≥2 (G(2,1,1)'C2).

The group G (d , 1, r )

The groupG(d,1,r) is the group of allr×r monomial matrices whose non-zero entries lie in µ_d.

G(d,1,r)'µdoSr.

The irreducible characters of G(d,1,r) are indexed by thed-partitions ofr. The field of definition ofG(d,1,r) isQ(ζd).

G(1,1,r)'A_r−1forr≥2,

G(2,1,r)'Br forr ≥2 (G(2,1,1)'C2).

The group G (d , 1, r )

The groupG(d,1,r) is the group of allr×r monomial matrices whose non-zero entries lie in µ_d.

G(d,1,r)'µdoSr.

The irreducible characters of G(d,1,r) are indexed by thed-partitions ofr. The field of definition ofG(d,1,r) isQ(ζd).

G(1,1,r)'A_r−1forr≥2,

G(2,1,r)'Br forr ≥2 (G(2,1,1)'C2).

The group G (d , 1, r )

The groupG(d,1,r) is the group of allr×r monomial matrices whose non-zero entries lie in µ_d.

G(d,1,r)'µdoSr.

The irreducible characters of G(d,1,r) are indexed by thed-partitions ofr. The field of definition ofG(d,1,r) isQ(ζd).

G(1,1,r)'A_r−1forr≥2, G(2,1,r)'Br forr ≥2

(G(2,1,1)'C2).

The group G (d , 1, r )

The groupG(d,1,r) is the group of allr×r monomial matrices whose non-zero entries lie in µ_d.

G(d,1,r)'µdoSr.

The irreducible characters of G(d,1,r) are indexed by thed-partitions ofr. The field of definition ofG(d,1,r) isQ(ζd).

G(1,1,r)'A_r−1forr≥2,

G(2,1,r)'Br forr ≥2 (G(2,1,1)'C2).

Ariki-Koike algebras

The“generic” Ariki-Koike algebraassociated toG(d,1,r) is the algebraH_d,r generated over the Laurent polynomial ring ind+ 1 indeterminates

Z[u0,u⁻¹₀ ,u1,u₁⁻¹, . . . ,u_d−1,u_d−1⁻¹ ,x,x⁻¹] by the elementss,t1,t2, . . . ,t_r−1 satisfying the relations

st1st1=t1st1s,

stj =tjs, for allj = 2, . . . ,r−1,

tj−1tjtj−1=tjtj−1tj, for allj = 2, . . . ,r−1, titj=tjti, for all 1≤i,j≤r−1 with|i−j|>1, (s−u₀)(s−u₁). . .(s−u_d−1) = 0,

(t_j−x)(t_j+ 1) = 0, for allj= 1, . . . ,r−1.

The Schur elements of Hd,r have been calculated independently by

Geck, Iancu, Malle (2000), Mathas (2004).

Let

φ:

uj7→ζ_d^jq^mj,(0≤j<d), x7→qⁿ

be a cyclotomic specialization forHd,r.

Proposition (C.)

The essential hyperplanes for G(d,1,r) are given by: N= 0.

kN+Ms−Mt= 0 for−r <k<r and 0≤s<t<d such that ζ_d^s −ζ_d^t is not a unit inZ[ζd].

The Schur elements of Hd,r have been calculated independently by Geck, Iancu, Malle (2000),

Mathas (2004).

Let

φ:

uj7→ζ_d^jq^mj,(0≤j<d), x7→qⁿ

be a cyclotomic specialization forHd,r.

Proposition (C.)

The essential hyperplanes for G(d,1,r) are given by: N= 0.

kN+Ms−Mt= 0 for−r <k<r and 0≤s<t<d such that ζ_d^s −ζ_d^t is not a unit inZ[ζd].