Rouquier Blocks for Cyclotomic Hecke Algebras

Rouquier Blocks for Cyclotomic Hecke Algebras

Michel Brou´ e

Institut Henri–Poincar´e

July 2007

After work of

Rapha¨el Rouquier, Gunter Malle, Michel B., Sungsoon Kim,

& Maria Chlouveraki

HECKE ALGEBRAS OF COMPLEX REFLECTION GROUPS

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

G

:

s

2

t

2

, G

:

s

3

t 3

and a field of realisation Q

(ring of integers Z

) : Q

= Q and Q

= Q (ζ

) .

The associated generic Hecke algebra is defined from such a presentation :

H(G

) := < S , T ;



 

 

STSTST = TSTSTS (S − u

)(S − u

) = 0 (T − v

)(T − v

) = 0

>

H(G

) := < S , T ;

( STS = TST

(S − u

)(S − u

)(S − u

) = 0 >

HECKE ALGEBRAS OF COMPLEX REFLECTION GROUPS Every complex reflection group G has a nice presentation “` a la Coxeter” :

G

:

s

2

t

2

, G

:

s

3

t 3

and a field of realisation Q

(ring of integers Z

) : Q

= Q and Q

= Q (ζ

) .

The associated generic Hecke algebra is defined from such a presentation :

H(G

) := < S , T ;



 

 

STSTST = TSTSTS (S − u

)(S − u

) = 0 (T − v

)(T − v

) = 0

>

H(G

) := < S , T ;

( STS = TST

(S − u

)(S − u

)(S − u

) = 0 >

HECKE ALGEBRAS OF COMPLEX REFLECTION GROUPS Every complex reflection group G has a nice presentation “` a la Coxeter” :

G

:

s

2

t

2

, G

:

s

3

t 3

and a field of realisation Q

(ring of integers Z

) : Q

= Q and Q

= Q (ζ

) .

The associated generic Hecke algebra is defined from such a presentation :

H(G

) := < S , T ;



 

 

STSTST = TSTSTS (S − u

)(S − u

) = 0 (T − v

)(T − v

) = 0

>

H(G

) := < S , T ;

( STS = TST

(S − u

)(S − u

)(S − u

) = 0 >

HECKE ALGEBRAS OF COMPLEX REFLECTION GROUPS Every complex reflection group G has a nice presentation “` a la Coxeter” :

G

:

s

2

t

2

, G

:

s

3

t 3

and a field of realisation Q

(ring of integers Z

) : Q

= Q and Q

= Q (ζ

) .

The associated generic Hecke algebra is defined from such a presentation :

H(G

) := < S , T ;



 

 

STSTST = TSTSTS (S − u

)(S − u

) = 0 (T − v

)(T − v

) = 0

>

(

1

The generic Hecke algebra H(G ) is free of rank |G | over the corresponding Laurent polynomial ring Z [(u

), (v

), . . . ].

2

It becomes a split semisimple algebra over a field obtained by extracting suitable roots of the indeterminates :

if G =

s

d m

t

e

· · · ,

then for

(x

= ζ

u

)

, (y

= ζ

v

)

I

the algebra Q

((x

), (y

), . . . ))H(G ) is split semisimple,

I

through the specialisation x

7→ 1 y

7→ 1, . . . , that algebra becomes the group algebra of G over Q

.

The above specialisation defines a bijection

Irr(G) −

→ Irr(H(G )) , χ 7→ χ

.

1

The generic Hecke algebra H(G ) is free of rank |G | over the corresponding Laurent polynomial ring Z [(u

), (v

), . . . ].

2

It becomes a split semisimple algebra over a field obtained by extracting suitable roots of the indeterminates :

if G =

s

d m

t

e

· · · ,

then for

(x

= ζ

u

)

, (y

= ζ

v

)

I

the algebra Q

((x

), (y

), . . . ))H(G ) is split semisimple,

I

through the specialisation x

7→ 1 y

7→ 1, . . . , that algebra becomes the group algebra of G over Q

.

The above specialisation defines a bijection

Irr(G) −

→ Irr(H(G )) , χ 7→ χ

.

1

The generic Hecke algebra H(G ) is free of rank |G | over the corresponding Laurent polynomial ring Z [(u

), (v

), . . . ].

2

It becomes a split semisimple algebra over a field obtained by extracting suitable roots of the indeterminates :

if G =

s

d m

t

e

· · · ,

then for

(x

= ζ

u

)

, (y

= ζ

v

)

I

the algebra Q

((x

), (y

), . . . ))H(G ) is split semisimple,

I

through the specialisation x

7→ 1 y

7→ 1, . . . , that algebra becomes the group algebra of G over Q

.

The above specialisation defines a bijection

Irr(G) −

→ Irr(H(G )) , χ 7→ χ

.

1

The generic Hecke algebra H(G ) is free of rank |G | over the corresponding Laurent polynomial ring Z [(u

), (v

), . . . ].

2

It becomes a split semisimple algebra over a field obtained by extracting suitable roots of the indeterminates :

if G =

s

d m

t

e

· · · ,

then for

(x

= ζ

u

)

, (y

= ζ

v

)

I

the algebra Q

((x

), (y

), . . . ))H(G ) is split semisimple,

I

through the specialisation x

7→ 1 y

7→ 1, . . . , that algebra becomes the group algebra of G over Q

.

The above specialisation defines a bijection

Irr(G) −

→ Irr(H(G )) , χ 7→ χ

.

1

The generic Hecke algebra H(G ) is free of rank |G | over the corresponding Laurent polynomial ring Z [(u

), (v

), . . . ].

2

It becomes a split semisimple algebra over a field obtained by extracting suitable roots of the indeterminates :

if G =

s

d m

t

e

· · · ,

then for

(x

= ζ

u

)

, (y

= ζ

v

)

I

the algebra Q

((x

), (y

), . . . ))H(G ) is split semisimple,

I

through the specialisation x

7→ 1 y

7→ 1, . . . , that algebra becomes the group algebra of G over Q

.

The above specialisation defines a bijection

4

The generic Hecke algebra H(G ) is endowed with a canonical symmetrizing form t : H(G ) → Z [(u

), (v

), . . . ]

I

which specialises to the canonical form of the group algebra Q

G ,

I

and satisfies some other condition.

5

The Schur elements of the irreducible characters of G are the elements s

∈ Z

[(x

), (y

), . . . ] defined by

t = X

χ∈Irr(G)

1 s

χ

.

4

The generic Hecke algebra H(G ) is endowed with a canonical symmetrizing form t : H(G ) → Z [(u

), (v

), . . . ]

I

which specialises to the canonical form of the group algebra Q

G ,

I

and satisfies some other condition.

5

The Schur elements of the irreducible characters of G are the elements s

∈ Z

[(x

), (y

), . . . ] defined by

t = X

χ∈Irr(G)

1 s

χ

.

4

The generic Hecke algebra H(G ) is endowed with a canonical symmetrizing form t : H(G ) → Z [(u

), (v

), . . . ]

I

which specialises to the canonical form of the group algebra Q

G ,

I

and satisfies some other condition.

5

The Schur elements of the irreducible characters of G are the elements s

∈ Z

[(x

), (y

), . . . ] defined by

t = X

χ∈Irr(G)

1 s

χ

.

4

The generic Hecke algebra H(G ) is endowed with a canonical symmetrizing form t : H(G ) → Z [(u

), (v

), . . . ]

I

which specialises to the canonical form of the group algebra Q

G ,

I

and satisfies some other condition.

5

The Schur elements of the irreducible characters of G are the elements s

∈ Z

[(x

), (y

), . . . ] defined by

t = X

χ∈Irr(G)

1

s

χ

.

Theorem (M. Chlouveraki) We have

s

(x) = ξ

N

Y

i∈I_χ

Ψ

(M

)

where

ξ

∈ Z

, N

is a degree zero monomial in Z

[x, x

], (n

)

are integers,

(Ψ

)

is a family of K -cyclotomic polynomials,

(M

)

is a family of degree zero primitive monomials in Z

[x, x

].

That factorisation is unique in K [x, x

] and the monomials (M

)

are

unique (up to inversion).

Schur elements of G

( x

= u

, x

= −u

y

= v

, y

= −v

Elements of the orbit under S(x

, x

) × S(y

, y

) of



 

 

 



 

 

 



s

:= Φ

(x

x

)Φ

(y

y

)Φ

(x

y

x

y

)Φ

(x

y

x

y

)) (corresponding to the four degree 1 irreducible characters)

s

:= 2Φ

(x

y

x

y

)Φ

(x

y

x

y

)

(corresponding to the two degree 2 irreducible characters)

Schur elements of G

( x

= u

, x

= −u

y

= v

, y

= −v

Elements of the orbit under S(x

, x

) × S(y

, y

) of



 

 

 



 

 

 



s

:= Φ

(x

x

)Φ

(y

y

)Φ

(x

y

x

y

)Φ

(x

y

x

y

)) (corresponding to the four degree 1 irreducible characters)

s

:= 2Φ

(x

y

x

y

)Φ

(x

y

x

y

)

(corresponding to the two degree 2 irreducible characters)

Schur elements of G

( x

= u

, x

= −u

y

= v

, y

= −v

Elements of the orbit under S(x

, x

) × S(y

, y

) of



 

 

 



 

 

 



s

:= Φ

(x

x

)Φ

(y

y

)Φ

(x

y

x

y

)Φ

(x

y

x

y

)) (corresponding to the four degree 1 irreducible characters)

s

:= 2Φ

(x

y

x

y

)Φ

(x

y

x

y

)

(corresponding to the two degree 2 irreducible characters)

Schur elements of G

:

x

:= u

x

:=

u

x

:=

u

Obtained by permutations of the parameters from 8

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> <

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> :

s

=

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

(corresponding to the three characters of degree 1)

s

=

x

x

Φ⁰₉(x1x₀⁻¹)

Φ

(x

x

)

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

(corresponding to the three characters of degree 2)

s

=

Φ

(x

x

x

) Φ

(x

x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

(corresponding to the unique character of degree 3)

Schur elements of G

: x

:= u

x

:=

u

x

:=

u

Obtained by permutations of the parameters from 8

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> <

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> :

s

=

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

(corresponding to the three characters of degree 1)

s

=

x

x

Φ⁰₉(x1x₀⁻¹)

Φ

(x

x

)

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

(corresponding to the three characters of degree 2)

s

=

Φ

(x

x

x

) Φ

(x

x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

(corresponding to the unique character of degree 3)

Schur elements of G

: x

:= u

x

:=

u

x

:=

u

Obtained by permutations of the parameters from 8

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> <

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

> >

>

s

=

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

(corresponding to the three characters of degree 1)

s

=

x

x

Φ⁰₉(x1x₀⁻¹)

Φ

(x

x

)

Φ

(x

x

)

Φ

(x

x

) Φ

(x

x

) Φ

(x

x

)

Φ

(x

x

x

) Φ

(x

x

x

)

(corresponding to the three characters of degree 2)

s

=

Φ

(x

x

x

) Φ

(x

x

x

)

2 −1 −1 0 2 −1 −1 00 2 −1 −1

Cyclotomic algebras

Let ζ be a root of unity. A ζ–cyclotomic specialisation of the generic Hecke algebra is a morphism

ϕ : x

7→ (ζ

q)

, y

7→ (ζ

q)

, . . . (m

, n

∈ Z ) , which gives rise to a ζ–cyclotomic Hecke algebra H

(G ).

A 1–cyclotomic Hecke algebra for G

=

s

2

t 2

:

< S , T ;



 

 

STSTST = TSTSTS (S − q

)(S + 1) = 0 (T − q )(T + 1) = 0



 

 

>

Cyclotomic algebras

Let ζ be a root of unity. A ζ–cyclotomic specialisation of the generic Hecke algebra is a morphism

ϕ : x

7→ (ζ

q)

, y

7→ (ζ

q)

, . . . (m

, n

∈ Z ) , which gives rise to a ζ–cyclotomic Hecke algebra H

(G ).

A 1–cyclotomic Hecke algebra for G

=

s

2

t 2

:

< S , T ;



 

 

STSTST = TSTSTS (S − q

)(S + 1) = 0 (T − q)(T + 1) = 0



 

 

>

A ζ

–cyclotomic Hecke algebra for B

(3) =

s

d

t 2

:

< S , T ;



 

 

STST = TSTS

(S − 1)(S − q)(S − q

) = 0 (T − q

)(T + 1) = 0



 

 

>

Relevance to character theory of finite reductive groups The unipotent characters in a given d –Harish–Chandra series

UnCh(G

; (L, λ)) are described by a suitable ζ

–cyclotomic Hecke algebra H

(L, λ)) for the corresponding d –cyclotomic Weyl group W

(L, λ)) :

UnCh(G

; (L, λ)) ←→ Irr(H

(L, λ))) .

A ζ

–cyclotomic Hecke algebra for B

(3) =

s

d

t 2

:

< S , T ;



 

 

STST = TSTS

(S − 1)(S − q)(S − q

) = 0 (T − q

)(T + 1) = 0



 

 

>

Relevance to character theory of finite reductive groups The unipotent characters in a given d –Harish–Chandra series

UnCh(G

; (L, λ)) are described by a suitable ζ

–cyclotomic Hecke algebra H

(L, λ)) for the corresponding d –cyclotomic Weyl group W

(L, λ)) :

UnCh(G

; (L, λ)) ←→ Irr(H

(L, λ))) .

ROUQUIER BLOCKS OF CYCLOTOMIC ALGEBRAS

• The Rouquier ring R

(q) is

R

(q ) := Z

[q, q

, ((q

− 1)

)

] . The Rouquier ring is a Dedekind domain.

• The Rouquier blocks of a cyclotomic algebra H

(G ) are the blocks (primitive central idempotents) of the algebra R

(q)H

(G ).

Let (e

), (e

)

be a pair of dual basis of H

(G ) relative to the canonical trace form t.

Then a Rouquier block b has the form

b = X

k



 X

χ∈b

χ

(e

) s



 e

where X

χ∈b

χ

(e

)

s

∈ R

(q)

ROUQUIER BLOCKS OF CYCLOTOMIC ALGEBRAS

• The Rouquier ring R

(q) is

R

(q ) := Z

[q, q

, ((q

− 1)

)

] . The Rouquier ring is a Dedekind domain.

• The Rouquier blocks of a cyclotomic algebra H

(G ) are the blocks (primitive central idempotents) of the algebra R

(q)H

(G ).

Let (e

), (e

)

be a pair of dual basis of H

(G ) relative to the canonical trace form t.

Then a Rouquier block b has the form

b = X

k



 X

χ∈b

χ

(e

) s



 e

where X

χ∈b

χ

(e

)

s

∈ R

(q)

ROUQUIER BLOCKS OF CYCLOTOMIC ALGEBRAS

• The Rouquier ring R

(q) is

R

(q ) := Z

[q, q

, ((q

− 1)

)

] . The Rouquier ring is a Dedekind domain.

• The Rouquier blocks of a cyclotomic algebra H

(G) are the blocks (primitive central idempotents) of the algebra R

(q)H

(G ).

Let (e

), (e

)

be a pair of dual basis of H

(G ) relative to the canonical trace form t.

Then a Rouquier block b has the form

b = X

k



 X

χ∈b

χ

(e

) s



 e

where X

χ∈b

χ

(e

)

s

∈ R

(q)

ROUQUIER BLOCKS OF CYCLOTOMIC ALGEBRAS

• The Rouquier ring R

(q) is

R

(q ) := Z

[q, q

, ((q

− 1)

)

] . The Rouquier ring is a Dedekind domain.

• The Rouquier blocks of a cyclotomic algebra H

(G) are the blocks (primitive central idempotents) of the algebra R

(q)H

(G ).

Let (e

), (e

)

be a pair of dual basis of H

(G ) relative to the canonical trace form t.

Then a Rouquier block b has the form

b = X



X χ

(e

)



e where X χ

(e

)

∈ R (q)

Relevance to character theory of finite reductive groups

For each integer d and each d –cuspidal pair (L, λ), let us identify UnCh(G

, (L, λ)) = Irr(H

(L, λ)) .

Then the trace of the Lusztig family partition

UnCh(G

) =

•

[

F ∈Fam(G^F)

F

on the d –Harish–Chandra series UnCh(G

, (L, λ)) is the partition into Rouquier blocks of Irr(H

(L, λ)) :

Irr(H(G

, (L, λ))) =

•

[

F ∈Fam(G^F)

F ∩ Irr(H

(L, λ))

| {z }

Rouquier Block or∅

.

Relevance to character theory of finite reductive groups For each integer d and each d –cuspidal pair (L, λ), let us identify

UnCh(G

, (L, λ)) = Irr(H

(L, λ)) .

Then the trace of the Lusztig family partition

UnCh(G

) =

•

[

F ∈Fam(G^F)

F

on the d –Harish–Chandra series UnCh(G

, (L, λ)) is the partition into Rouquier blocks of Irr(H

(L, λ)) :

Irr(H(G

, (L, λ))) =

•

[

F ∈Fam(G^F)

F ∩ Irr(H

(L, λ))

| {z }

Rouquier Block or∅

.

Relevance to character theory of finite reductive groups For each integer d and each d –cuspidal pair (L, λ), let us identify

UnCh(G

, (L, λ)) = Irr(H

(L, λ)) . Then the trace of the Lusztig family partition

UnCh(G

) =

•

[

F ∈Fam(G^F)

F

on the d –Harish–Chandra series UnCh(G

, (L, λ)) is the partition into Rouquier blocks of Irr(H

(L, λ)) :

Irr(H(G

, (L, λ))) =

•

[

F ∈Fam(G^F)

F ∩ Irr(H

(L, λ))

| {z }

Rouquier Block or∅

.

Relevance to character theory of finite reductive groups For each integer d and each d –cuspidal pair (L, λ), let us identify

UnCh(G

, (L, λ)) = Irr(H

(L, λ)) . Then the trace of the Lusztig family partition

UnCh(G

) =

•

[

F ∈Fam(G^F)

F

on the d –Harish–Chandra series UnCh(G

, (L, λ)) is the partition into Rouquier blocks of Irr(H

(L, λ)) :

Irr(H(G

, (L, λ))) =

•

[

F ∈Fam(G^F)

F ∩ Irr(H

(L, λ))

| {z }

Rouquier Block or∅

.

Computation of Rouquier Blocks

p–Blocks and Bad primes

For p a prime ideal of Z

, the blocks of the localised algebra H(G )

(resp. H

(G )

) are called p–blocks of H(G ) (resp. H

(G )).

A bad prime for H

(G ) is a prime ideal of Z

which divides some Schur element s

.

A prime p is bad if and only if there is at least one p–block which is nontrivial.

Proposition

The Rouquier blocks of H

(G ) are the union of the p–blocks of H

(G )

for p running over the set bad(H

(G )) of bad primes.

Computation of Rouquier Blocks p–Blocks and Bad primes

For p a prime ideal of Z

, the blocks of the localised algebra H(G )

(resp. H

(G )

) are called p–blocks of H(G ) (resp. H

(G )).

A bad prime for H

(G ) is a prime ideal of Z

which divides some Schur element s

.

A prime p is bad if and only if there is at least one p–block which is nontrivial.

Proposition

The Rouquier blocks of H

(G ) are the union of the p–blocks of H

(G )

for p running over the set bad(H

(G )) of bad primes.

Computation of Rouquier Blocks

p–Blocks and Bad primes

For p a prime ideal of Z

, the blocks of the localised algebra H(G )

(resp. H

(G )

) are called p–blocks of H(G ) (resp. H

(G )).

A bad prime for H

(G ) is a prime ideal of Z

which divides some Schur element s

.

A prime p is bad if and only if there is at least one p–block which is nontrivial.

Proposition

The Rouquier blocks of H

(G ) are the union of the p–blocks of H

(G )

for p running over the set bad(H

(G )) of bad primes.

Computation of Rouquier Blocks

p–Blocks and Bad primes

For p a prime ideal of Z

, the blocks of the localised algebra H(G )

(resp. H

(G )

) are called p–blocks of H(G ) (resp. H

(G )).

A bad prime for H

(G ) is a prime ideal of Z

which divides some Schur element s

.

A prime p is bad if and only if there is at least one p–block which is nontrivial.

Proposition

The Rouquier blocks of H

(G ) are the union of the p–blocks of H

(G )

for p running over the set bad(H

(G )) of bad primes.

Computation of Rouquier Blocks

p–Blocks and Bad primes

For p a prime ideal of Z

, the blocks of the localised algebra H(G )

(resp. H

(G )

) are called p–blocks of H(G ) (resp. H

(G )).

A bad prime for H

(G ) is a prime ideal of Z

which divides some Schur element s

.

A prime p is bad if and only if there is at least one p–block which is nontrivial.

Proposition

The Rouquier blocks of H

(G ) are the union of the p–blocks of H

(G )

for p running over the set bad(H

(G )) of bad primes.

Computation of Rouquier Blocks

p–Blocks and Bad primes

For p a prime ideal of Z

, the blocks of the localised algebra H(G )

(resp. H

(G )

) are called p–blocks of H(G ) (resp. H

(G )).

A bad prime for H

(G ) is a prime ideal of Z

which divides some Schur element s

.

A prime p is bad if and only if there is at least one p–block which is nontrivial.

Proposition

The Rouquier blocks of H

(G ) are the union of the p–blocks of H

(G )

for p running over the set bad(H

(G )) of bad primes.

p–Blocks and p–essential monomials

Definition Let M = Q

i

x

y

· · · be a monomial.

We say that M is p–essential if there exist a Q

–cyclotomic polynomial Ψ and an irreducible character χ of G such that

1

Ψ(M ) divides s

,

2

p divides Ψ(1).

A character χ as above is then said to be p–singular at M. If M is p–essential, the hyperplane H

defined by Log(M ), i.e., P

i

a

X

+ P

j

b

Y

+ · · · = 0 , is called a p–essential hyperplane.

p–Blocks and p–essential monomials

Definition Let M = Q

i

x

y

· · · be a monomial.

We say that M is p–essential if there exist a Q

–cyclotomic polynomial Ψ and an irreducible character χ of G such that

1

Ψ(M ) divides s

,

2

p divides Ψ(1).

A character χ as above is then said to be p–singular at M. If M is p–essential, the hyperplane H

defined by Log(M ), i.e., P

i

a

X

+ P

j

b

Y

+ · · · = 0 , is called a p–essential hyperplane.

p–Blocks and p–essential monomials

Definition Let M = Q

i

x

y

· · · be a monomial.

We say that M is p–essential if there exist a Q

–cyclotomic polynomial Ψ and an irreducible character χ of G such that

1

Ψ(M ) divides s

,

2

p divides Ψ(1).

A character χ as above is then said to be p–singular at M.

If M is p–essential, the hyperplane H

defined by Log(M ), i.e., P

i

a

X

+ P

j

b

Y

+ · · · = 0 , is called a p–essential hyperplane.

p–Blocks and p–essential monomials

Definition Let M = Q

i

x

y

· · · be a monomial.

We say that M is p–essential if there exist a Q

–cyclotomic polynomial Ψ and an irreducible character χ of G such that

1

Ψ(M ) divides s

,

2

p divides Ψ(1).

A character χ as above is then said to be p–singular at M.

If M is p–essential, the hyperplane H

defined by Log(M ), i.e., P

i

a

X

+ P

j

b

Y

+ · · · = 0 , is called a p–essential hyperplane.

p–Blocks and p–essential monomials

Definition Let M = Q

i

x

y

· · · be a monomial.

We say that M is p–essential if there exist a Q

–cyclotomic polynomial Ψ and an irreducible character χ of G such that

1

Ψ(M ) divides s

,

2

p divides Ψ(1).

A character χ as above is then said to be p–singular at M.

If M is p–essential, the hyperplane H

defined by Log(M ), i.e., P

i

a

X

+ P

j

b

Y

+ · · · = 0 , is called a p–essential hyperplane.

p–Blocks and p–essential monomials

Definition Let M = Q

i

x

y

· · · be a monomial.

We say that M is p–essential if there exist a Q

–cyclotomic polynomial Ψ and an irreducible character χ of G such that

1

Ψ(M ) divides s

,

2

p divides Ψ(1).

A character χ as above is then said to be p–singular at M.

If M is p–essential, the hyperplane H

defined by Log(M ), i.e., P

i

a

X

+ P

j

b

Y

+ · · · = 0 , is called a p–essential hyperplane.

p–Blocks and p–essential monomials

Definition Let M = Q

i

x

y

· · · be a monomial.

We say that M is p–essential if there exist a Q

–cyclotomic polynomial Ψ and an irreducible character χ of G such that

1

Ψ(M ) divides s

,

2

p divides Ψ(1).

A character χ as above is then said to be p–singular at M.

If M is p–essential, the hyperplane H

defined by Log(M ), i.e., P

i

a

X

+ P

j

b

Y

+ · · · = 0 , is called a p–essential hyperplane.

p–Blocks and p–essential monomials

Definition Let M = Q

i

x

y

· · · be a monomial.

We say that M is p–essential if there exist a Q

–cyclotomic polynomial Ψ and an irreducible character χ of G such that

1

Ψ(M ) divides s

,

2

p divides Ψ(1).

A character χ as above is then said to be p–singular at M.

If M is p–essential, the hyperplane H

defined by Log(M ), i.e., P

i

a

X

+ P

j

b

Y

+ · · · = 0 , is called a p–essential hyperplane.

Cyclotomic specialisation and p–singularity

Definition

Let ϕ be a cyclotomic specialisation of H(G ), defined by the family of integers (m

), (n

), . . . . Let M be a p–essential monomial.

We say that ϕ is p–singular at M (or at H

) if ϕ(M ) = 1 ,

i.e., if the parameters of ϕ belong to the p–essential hyperplane H

: X

i

a

m

+ X

j

b

n

+ · · · = 0 .

Cyclotomic specialisation and p–singularity

Definition

Let ϕ be a cyclotomic specialisation of H(G ), defined by the family of integers (m

), (n

), . . . . Let M be a p–essential monomial.

We say that ϕ is p–singular at M (or at H

) if ϕ(M ) = 1 ,

i.e., if the parameters of ϕ belong to the p–essential hyperplane H

: X

i

a

m

+ X

j

b

n

+ · · · = 0 .