Cyclotomic Root Systems

(1)

Cyclotomic Root Systems

Michel Brou´ e

Universit´e Paris-Diderot Paris 7

En hommage ` a Serge Bouc – mais sans cat´ egories ni foncteurs

—

Variations on an old work of Gabi Nebe A joint work with Ruth Corran and Jean Michel

(2)

A few words about motivation

Spetses

(Gunter Malle, Jean Michel, Michel B., now also Olivier Dudas, C´edric Bonnaf´e)

A finite reductive groupG^F is determined by the choice ofGandF, and that choice is in turn determined by the choice of aroot datum(X,R,Y,R^∨), a finite order automorphismφof the root datum, and the basic fieldFq.

Some data concerning G^F, such as the parametrization of unipotent characters, their generic degrees, Frobenius eigenvalues, and also the families and their Fourier matrices, depend ONLYon theQ-representation of the Weyl group attached to the root datum, and on the automorphism φ.

This may be surprising: think of Sp2r(q) and SO2r+1(q).

More data concerningG^F, such as the parametrization of unipotent classes, the values of unipotent characters on unipotent elements, depend on the entire root datum.

Michel Brou´e Cyclotomic Root Systems

(3)

A few words about motivation

Spetses

(Gunter Malle, Jean Michel, Michel B., now also Olivier Dudas, C´edric Bonnaf´e) A finite reductive groupG^F is determined by the choice ofGandF, and that choice is in turn determined by the choice of aroot datum(X,R,Y,R^∨), a finite order automorphismφof the root datum, and the basic fieldFq.

(4)

A few words about motivation

Spetses

(Gunter Malle, Jean Michel, Michel B., now also Olivier Dudas, C´edric Bonnaf´e)

A finite reductive groupG^F is determined by the choice ofGandF, and that choice is in turn determined by the choice of aroot datum(X,R,Y,R^∨), a finite order automorphismφof the root datum, and the basic fieldFq.

(5)

A few words about motivation

Spetses

(6)

A few words about motivation

Spetses

(7)

A few words about motivation

Spetses

(8)

A few words about motivation

Spetses

(9)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R).

On can classify the root systems R and then classify the Weyl groups

G (R). Or proceed the opposite way: it’s what we did for cyclotomic root

systems and their (cyclotomic) Weyl groups.

(10)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R). On can classify the root systems R and then classify the Weyl groups G (R). Or proceed the opposite way: it’s what we did for cyclotomic root systems and their (cyclotomic) Weyl groups.

(11)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors.

We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R).

On can classify the root systems R and then classify the Weyl groups

G (R). Or proceed the opposite way: it’s what we did for cyclotomic root

systems and their (cyclotomic) Weyl groups.

(12)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R). On can classify the root systems R and then classify the Weyl groups G (R). Or proceed the opposite way: it’s what we did for cyclotomic root systems and their (cyclotomic) Weyl groups.

(13)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R).

On can classify the root systems R and then classify the Weyl groups

G (R). Or proceed the opposite way: it’s what we did for cyclotomic root

systems and their (cyclotomic) Weyl groups.

(14)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R). On can classify the root systems R and then classify the Weyl groups G (R). Or proceed the opposite way: it’s what we did for cyclotomic root systems and their (cyclotomic) Weyl groups.

(15)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R).

On can classify the root systems R and then classify the Weyl groups

G (R). Or proceed the opposite way: it’s what we did for cyclotomic root

systems and their (cyclotomic) Weyl groups.

(16)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R).

On can classify the root systems R and then classify the Weyl groups G (R). Or proceed the opposite way: it’s what we did for cyclotomic root systems and their (cyclotomic) Weyl groups.

(17)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R).

On can classify the root systems R and then classify the Weyl groups G (R).

Or proceed the opposite way: it’s what we did for cyclotomic root

systems and their (cyclotomic) Weyl groups.

(18)

Root systems and Weyl groups: Bourbaki’s definition

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

The group generated by all reflections s

_α,α^∨

is the Weyl group G (R).

On can classify the root systems R and then classify the Weyl groups G (R). Or proceed the opposite way: it’s what we did for cyclotomic root systems and their (cyclotomic) Weyl groups.

(19)

Complex (or, rather, cyclotomic) reflection groups

Let k be a subfield of

C

which is stable under complex conjugation.

Let V and W be finite dimensional k -vector spaces endowed with a hermitian duality V × W → k .

A reflection s on V is defined by a triple (L

_s

, M

_s

, ζ

_s

) where

ζ

s

∈

µ(k),

L

s

is a line in V and M

s

is a line in W such that hL

_s

, M

s

i 6= 0

and s is the automorphism of V defined by s (v) = v − hv, α

^∨

iα

whenever α ∈ L

s

and α

^∨

∈ M

s

are such that hα, α

^∨

i = 1 − ζ

s

.

(20)

Complex (or, rather, cyclotomic) reflection groups

Let k be a subfield of

C

which is stable under complex conjugation.

Let V and W be finite dimensional k -vector spaces endowed with a hermitian duality V × W → k .

A reflection s on V is defined by a triple (L

_s

, M

_s

, ζ

_s

) where

ζ

s

∈

µ(k),

L

s

is a line in V and M

s

is a line in W such that hL

_s

, M

s

i 6= 0 and s is the automorphism of V defined by

s (v) = v − hv, α

^∨

iα

whenever α ∈ L

s

and α

^∨

∈ M

s

are such that hα, α

^∨

i = 1 − ζ

s

.

(21)

Complex (or, rather, cyclotomic) reflection groups

Let k be a subfield of

C

which is stable under complex conjugation.

Let V and W be finite dimensional k -vector spaces endowed with a hermitian duality V × W → k .

A reflection s on V is defined by a triple (L

_s

, M

_s

, ζ

_s

) where ζ

s

∈

µ(k),

L

s

is a line in V and M

s

is a line in W such that hL

_s

, M

s

i 6= 0 and s is the automorphism of V defined by

s (v) = v − hv, α

^∨

iα

whenever α ∈ L

s

and α

^∨

∈ M

s

are such that hα, α

^∨

i = 1 − ζ

s

.

(22)

Complex (or, rather, cyclotomic) reflection groups

Let k be a subfield of

C

which is stable under complex conjugation.

Let V and W be finite dimensional k -vector spaces endowed with a hermitian duality V × W → k .

A reflection s on V is defined by a triple (L

_s

, M

_s

, ζ

_s

) where ζ

s

∈

µ(k),

L

s

is a line in V and M

s

is a line in W such that hL

_s

, M

s

i 6= 0

and s is the automorphism of V defined by s (v) = v − hv, α

^∨

iα

whenever α ∈ L

s

and α

^∨

∈ M

s

are such that hα, α

^∨

i = 1 − ζ

s

.

(23)

Complex (or, rather, cyclotomic) reflection groups

Let k be a subfield of

C

which is stable under complex conjugation.

Let V and W be finite dimensional k -vector spaces endowed with a hermitian duality V × W → k .

A reflection s on V is defined by a triple (L

_s

, M

_s

, ζ

_s

) where ζ

s

∈

µ(k),

L

s

is a line in V and M

s

is a line in W such that hL

_s

, M

s

i 6= 0 and s is the automorphism of V defined by

s (v) = v − hv, α

^∨

iα

whenever α ∈ L

s

and α

^∨

∈ M

s

are such that hα, α

^∨

i = 1 − ζ

s

.

(24)

Shephard–Todd groups and their fields

Irreducible cyclotomic reflection groups are classified.

An infinite series G (de, e , r) for d , e , r ∈

N

− {0}, G (de, e, r ) ⊂ GL

_r

(

Q

(ζ

_de

)) is irreducible

→except ford=e=r = 1 or 2.

Its field of definition is

Q

(ζ

_de

),

→except ford= 1 andr = 2 where it is the real fieldQ(ζ_e+ζ_e⁻¹).

The ring of integers ofQ(ζn) is Z[ζn].

In general (for example ifn>90, but also for other values ofnbetween 22 and 90) it isnota principal ideal domain.

34 exceptional irreducible groups in dimension 2 to 8.

Their field of definition (all subfields of “small” cyclotomic fields) have the remarkable property that all their rings of integers are principal ideal domains.

(25)

Shephard–Todd groups and their fields

Irreducible cyclotomic reflection groups are classified.

An infinite series G (de, e , r) for d , e , r ∈

N

− {0}, G (de, e , r) ⊂ GL

_r

(

Q

(ζ

_de

)) is irreducible

Its field of definition is

Q

(ζ

_de

),

The ring of integers ofQ(ζ_n) is Z[ζ_n].

34 exceptional irreducible groups in dimension 2 to 8.

(26)

Shephard–Todd groups and their fields

Irreducible cyclotomic reflection groups are classified.

An infinite series G (de, e , r) for d , e , r ∈

N

− {0}, G (de, e , r) ⊂ GL

_r

(

Q

(ζ

_de

)) is irreducible

Its field of definition is

Q

(ζ

_de

),

34 exceptional irreducible groups in dimension 2 to 8.

(27)

Shephard–Todd groups and their fields

Irreducible cyclotomic reflection groups are classified.

An infinite series G (de, e , r) for d , e , r ∈

N

− {0}, G (de, e , r) ⊂ GL

_r

(

Q

(ζ

_de

)) is irreducible

Its field of definition is

Q

(ζ

_de

),

→except ford= 1 andr = 2 where it is the real fieldQ(ζ_e+ζ_e⁻¹). The ring of integers ofQ(ζ_n) is Z[ζ_n].

34 exceptional irreducible groups in dimension 2 to 8.

(28)

Shephard–Todd groups and their fields

Irreducible cyclotomic reflection groups are classified.

An infinite series G (de, e , r) for d , e , r ∈

N

− {0}, G (de, e , r) ⊂ GL

_r

(

Q

(ζ

_de

)) is irreducible

Its field of definition is

Q

(ζ

_de

),

34 exceptional irreducible groups in dimension 2 to 8.

(29)

Shephard–Todd groups and their fields

Irreducible cyclotomic reflection groups are classified.

An infinite series G (de, e , r) for d , e , r ∈

N

− {0}, G (de, e , r) ⊂ GL

_r

(

Q

(ζ

_de

)) is irreducible

Its field of definition is

Q

(ζ

_de

),

The ring of integers ofQ(ζ_n) isZ[ζ_n].

34 exceptional irreducible groups in dimension 2 to 8.

(30)

Shephard–Todd groups and their fields

Irreducible cyclotomic reflection groups are classified.

An infinite series G (de, e , r) for d , e , r ∈

N

− {0}, G (de, e , r) ⊂ GL

_r

(

Q

(ζ

_de

)) is irreducible

Its field of definition is

Q

(ζ

_de

),

The ring of integers ofQ(ζ_n) isZ[ζ_n].

34 exceptional irreducible groups in dimension 2 to 8.

(31)

Ordinary Root systems

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors. We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

_α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

(32)

Ordinary Root systems

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors.

We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

(33)

Ordinary Root systems

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors.

We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection s

α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

(34)

Ordinary Root systems

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors.

We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

(35)

Ordinary Root systems

Let V and W be finite dimensional

Q

-vector spaces endowed with a duality V × W →

Q

.

Let R := {(α, α

^∨

)} ⊂ V × W be a family of nonzero vectors.

We say that R is a root system if

(RS1)

R is finite and its projection on V generates V ,

(RS2)

for all (α, α

^∨

) ∈ R, hα, α

^∨

i = 2 and the reflection

s

α,α^∨

: v 7→ v − hv , α

^∨

iα stabilizes R,

(RS3)

for all (α, α

^∨

), (β, β

^∨

) ∈ R, hα, β

^∨

i ∈

Z

.

(36)

Z

^k

–Root Systems

It is a set of triples R = {r = (I

_r

, J

_r

, ζ

_r

)} where

ζ

_r

∈

µ(k),

I

_r

is a rank one

Zk

-submodule of V , and J

_r

is a rank one

Zk

-submodule of W ,

such that

(RS1)

the family (I

r

) generates V ,

(RS2)

hI

_r

, J

_r

i = [1 − ζ

_r

] (the principal ideal generated by 1 − ζ

_r

), and if

P

i

hα

_i

, β

_i

i = 1 − ζ

_r

, then the reflection s

_r

: v 7→ v −

X

i

hv , β

_i

iα

_i

stabilizes R,

(RS3)

Whenever

r,r⁰

∈ R, hI

_r

, J

_r⁰

i ⊂

Zk

.

(37)

Z

^k

–Root Systems

It is a set of triples R = {r = (I

_r

, J

_r

, ζ

_r

)} where ζ

_r

∈

µ(k),

I

_r

is a rank one

Zk

-submodule of V , and J

_r

is a rank one

Zk

-submodule of W ,

such that

(RS1)

the family (I

r

) generates V ,

(RS2)

hI

_r

, J

_r

i = [1 − ζ

_r

] (the principal ideal generated by 1 − ζ

_r

), and if

P

i

hα

_i

, β

_i

i = 1 − ζ

_r

, then the reflection s

_r

: v 7→ v −

X

i

hv , β

_i

iα

_i

stabilizes R,

(RS3)

Whenever

r,r⁰

∈ R, hI

_r

, J

_r⁰

i ⊂

Zk

.

(38)

Z

^k

–Root Systems

It is a set of triples R = {r = (I

_r

, J

_r

, ζ

_r

)} where ζ

_r

∈

µ(k),

I

_r

is a rank one

Zk

-submodule of V , and J

_r

is a rank one

Zk

-submodule of W ,

such that

(RS1)

the family (I

r

) generates V ,

(RS2)

hI

_r

, J

_r

i = [1 − ζ

_r

] (the principal ideal generated by 1 − ζ

_r

), and if

P

i

hα

_i

, β

_i

i = 1 − ζ

_r

, then the reflection s

_r

: v 7→ v −

X

i

hv , β

_i

iα

_i

stabilizes R,

(RS3)

Whenever

r,r⁰

∈ R, hI

_r

, J

_r⁰

i ⊂

Zk

.

(39)

Z

^k

–Root Systems

It is a set of triples R = {r = (I

_r

, J

_r

, ζ

_r

)} where ζ

_r

∈

µ(k),

I

_r

is a rank one

Zk

-submodule of V , and J

_r

is a rank one

Zk

-submodule of W ,

such that

(RS1)

the family (I

r

) generates V ,

(RS2)

hI

_r

, J

_r

i = [1 − ζ

_r

] (the principal ideal generated by 1 − ζ

_r

), and if

P

i

hα

_i

, β

_i

i = 1 − ζ

_r

, then the reflection s

_r

: v 7→ v −

X

i

hv , β

_i

iα

_i

stabilizes R,

(RS3)

Whenever

r,r⁰

∈ R, hI

_r

, J

_r⁰

i ⊂

Zk

.

(40)

Z

^k

–Root Systems

It is a set of triples R = {r = (I

_r

, J

_r

, ζ

_r

)} where ζ

_r

∈

µ(k),

I

_r

is a rank one

Zk

-submodule of V , and J

_r

is a rank one

Zk

-submodule of W ,

such that

(RS1)

the family (I

r

) generates V ,

(RS2)

hI

_r

, J

_r

i = [1 − ζ

_r

] (the principal ideal generated by 1 − ζ

_r

), and if

P

i

hα

_i

, β

_i

i = 1 − ζ

_r

, then the reflection s

_r

: v 7→ v −

X

i

hv , β

_i

iα

_i

stabilizes R,

(RS3)

Whenever

r,r⁰

∈ R, hI

_r

, J

_r⁰

i ⊂

Zk

.

(41)

Z

^k

–Root Systems

It is a set of triples R = {r = (I

_r

, J

_r

, ζ

_r

)} where ζ

_r

∈

µ(k),

I

_r

is a rank one

Zk

-submodule of V , and J

_r

is a rank one

Zk

-submodule of W ,

such that

(RS1)

the family (I

r

) generates V ,

(RS2)

hI

_r

, J

_r

i = [1 − ζ

_r

] (the principal ideal generated by 1 − ζ

_r

), and if

P

i

hα

_i

, β

_i

i = 1 − ζ

_r

, then the reflection s

_r

: v 7→ v −

X

i

hv , β

_i

iα

_i

stabilizes R,

(RS3)

Whenever

r,r⁰

∈ R, hI

_r

, J

_r⁰

i ⊂

Zk

.

(42)

Z

^k

–Root Systems

It is a set of triples R = {r = (I

_r

, J

_r

, ζ

_r

)} where ζ

_r

∈

µ(k),

I

_r

is a rank one

Zk

-submodule of V , and J

_r

is a rank one

Zk

-submodule of W ,

such that

(RS1)

the family (I

r

) generates V ,

(RS2)

hI

_r

, J

_r

i = [1 − ζ

_r

] (the principal ideal generated by 1 − ζ

_r

), and if

P

i

hα

_i

, β

_i

i = 1 − ζ

_r

, then the reflection s

_r

: v 7→ v −

X

i

hv , β

_i

iα

_i

stabilizes R,

(RS3)

Whenever

r,r⁰

∈ R, hI

_r

, J

_r⁰

i ⊂

Zk

.

(43)

Theorem

(1)

G (R) := hs

_r

i

_r∈R

is finite and V

^G^(R)

= 0.

(2)

Conversely, whenever G is a finite subgroup of GL(V ) generated by reflections such that V

^G

= 0, there exists a

Zk

-root system R such that G = G (R).

(3)

Arr(G (R)) = Arr(R) . Parabolic subsystems

Let F be a face (intersection of reflecting hyperplanes) of R (or of G(R)). Then the fixator G (R)

_F

is called a parabolic subgroup of G (R).

If V

_F

denotes the sum of reflecting lines of G(R)

_F

, we have V

_F^G^(R)^F

= 0.

Then R

_F

:= {r | s

_r

∈ G (R)

_F

} is a

Zk

root system in V

_F

.

(44)

Theorem

(1)

G (R) := hs

_r

i

_r∈R

is finite and V

^G^(R)

= 0.

(2)

Conversely, whenever G is a finite subgroup of GL(V ) generated by reflections such that V

^G

= 0, there exists a

Zk

-root system R such that G = G (R).

(3)

Arr(G (R)) = Arr(R) . Parabolic subsystems

Let F be a face (intersection of reflecting hyperplanes) of R (or of G(R)). Then the fixator G (R)

_F

is called a parabolic subgroup of G (R).

If V

_F

denotes the sum of reflecting lines of G(R)

_F

, we have V

_F^G^(R)^F

= 0. Then R

_F

:= {r | s

_r

∈ G (R)

_F

} is a

Zk

root system in V

_F

.

(45)

Theorem

(1)

G (R) := hs

_r

i

_r∈R

is finite and V

^G^(R)

= 0.

(2)

Conversely, whenever G is a finite subgroup of GL(V ) generated by reflections such that V

^G

= 0, there exists a

Zk

-root system R such that G = G (R).

(3)

Arr(G (R)) = Arr(R) . Parabolic subsystems

Let F be a face (intersection of reflecting hyperplanes) of R (or of G(R)). Then the fixator G (R)

_F

is called a parabolic subgroup of G (R).

If V

_F

denotes the sum of reflecting lines of G(R)

_F

, we have V

_F^G^(R)^F

= 0.

Then R

_F

:= {r | s

_r

∈ G (R)

_F

} is a

Zk

root system in V

_F

.

(46)

Theorem

(1)

G (R) := hs

_r

i

_r∈R

is finite and V

^G^(R)

= 0.

(2)

Conversely, whenever G is a finite subgroup of GL(V ) generated by reflections such that V

^G

= 0, there exists a

Zk

-root system R such that G = G (R).

(3)

Arr(G (R)) = Arr(R) .

Parabolic subsystems

Let F be a face (intersection of reflecting hyperplanes) of R (or of G(R)). Then the fixator G (R)

_F

is called a parabolic subgroup of G (R).

If V

_F

denotes the sum of reflecting lines of G(R)

_F

, we have V

_F^G^(R)^F

= 0. Then R

_F

:= {r | s

_r

∈ G (R)

_F

} is a

Zk

root system in V

_F

.

(47)

Theorem

(1)

G (R) := hs

_r

i

_r∈R

is finite and V

^G^(R)

= 0.

(2)

Conversely, whenever G is a finite subgroup of GL(V ) generated by reflections such that V

^G

= 0, there exists a

Zk

-root system R such that G = G (R).

(3)

Arr(G (R)) = Arr(R) .

Parabolic subsystems

Let F be a face (intersection of reflecting hyperplanes) of R (or of G(R)).

Then the fixator G (R)

_F

is called a parabolic subgroup of G (R).

If V

_F

denotes the sum of reflecting lines of G(R)

_F

, we have V

_F^G^(R)^F

= 0.

Then R

_F

:= {r | s

_r

∈ G (R)

_F

} is a

Zk

root system in V

_F

.

(48)

Theorem

(1)

G (R) := hs

_r

i

_r∈R

is finite and V

^G^(R)

= 0.

(2)

Conversely, whenever G is a finite subgroup of GL(V ) generated by reflections such that V

^G

= 0, there exists a

Zk

-root system R such that G = G (R).

(3)

Arr(G (R)) = Arr(R) .

Parabolic subsystems

Let F be a face (intersection of reflecting hyperplanes) of R (or of G(R)).

Then the fixator G (R)

_F

is called a parabolic subgroup of G (R).

If V

_F

denotes the sum of reflecting lines of G(R)

_F

, we have V

_F^G^(R)^F

= 0. Then R

_F

:= {r | s

_r

∈ G (R)

_F

} is a

Zk

root system in V

_F

.

(49)

Theorem

(1)

G (R) := hs

_r

i

_r∈R

is finite and V

^G^(R)

= 0.

(2)

Conversely, whenever G is a finite subgroup of GL(V ) generated by reflections such that V

^G

= 0, there exists a

Zk

-root system R such that G = G (R).

(3)

Arr(G (R)) = Arr(R) .

Parabolic subsystems

Let F be a face (intersection of reflecting hyperplanes) of R (or of G(R)).

Then the fixator G (R)

_F

is called a parabolic subgroup of G (R).

If V

_F

denotes the sum of reflecting lines of G(R)

_F

, we have V

_F^G^(R)^F

= 0.

Then R

_F

:= {r | s

_r

∈ G (R)

_F

} is a

Zk

root system in V

_F

.

(50)

Theorem

(1)

G (R) := hs

_r

i

_r∈R

is finite and V

^G^(R)

= 0.

(2)

Conversely, whenever G is a finite subgroup of GL(V ) generated by reflections such that V

^G

= 0, there exists a

Zk

-root system R such that G = G (R).

(3)

Arr(G (R)) = Arr(R) .

Parabolic subsystems

Let F be a face (intersection of reflecting hyperplanes) of R (or of G(R)).

Then the fixator G (R)

_F

is called a parabolic subgroup of G (R).

If V

_F

denotes the sum of reflecting lines of G(R)

_F

, we have V

_F^G^(R)^F

= 0.

Then R

_F

:= {r | s

_r

∈ G (R)

_F

} is a

Zk

root system in V

_F

.

(51)

Genera, Root and Weight Lattices

GL(V ) acts on root systems :

g · (I , J, ζ ) := (g (I ), g

^∨

(J), ζ ) .

If

a

is a fractional ideal in k, we set

a

· (I, J, ζ) := (aI,

a^−∗

J, ζ) . Root lattices, weight lattices:

Q

R

:=

X

r∈R

I

r

and Q

R^∨

:=

X

r∈R

J

r

P

R

:= {x ∈ V | ∀y ∈ Q

R^∨

, hx, yi ∈

Zk

} and P

R^∨

:= ... There is a Aut(R)/G(R)-invariant natural pairing

(P

R

/Q

R

) × (P

R^∨

/Q

R^∨

) → k/

Zk

.

(52)

Genera, Root and Weight Lattices

GL(V ) acts on root systems :

g · (I , J, ζ ) := (g (I ), g

^∨

(J), ζ ) . If

a

is a fractional ideal in k, we set

a

· (I, J, ζ) := (aI,

a^−∗

J, ζ) .

Root lattices, weight lattices: Q

R

:=

X

r∈R

I

r

and Q

R^∨

:=

X

r∈R

J

r

P

R

:= {x ∈ V | ∀y ∈ Q

R^∨

, hx, yi ∈

Zk

} and P

R^∨

:= ... There is a Aut(R)/G(R)-invariant natural pairing

(P

R

/Q

R

) × (P

R^∨

/Q

R^∨

) → k/

Zk

.

(53)

Genera, Root and Weight Lattices

GL(V ) acts on root systems :

g · (I , J, ζ ) := (g (I ), g

^∨

(J), ζ ) . If

a

is a fractional ideal in k, we set

a

· (I, J, ζ) := (aI,

a^−∗

J, ζ) . Root lattices, weight lattices:

Q

R

:=

X

r∈R

I

r

and Q

R^∨

:=

X

r∈R

J

r

P

R

:= {x ∈ V | ∀y ∈ Q

R^∨

, hx, yi ∈

Zk

} and P

R^∨

:= ... There is a Aut(R)/G(R)-invariant natural pairing

(P

R

/Q

R

) × (P

R^∨

/Q

R^∨

) → k/

Zk

.

(54)

Genera, Root and Weight Lattices

GL(V ) acts on root systems :

g · (I , J, ζ ) := (g (I ), g

^∨

(J), ζ ) . If

a

is a fractional ideal in k, we set

a

· (I, J, ζ) := (aI,

a^−∗

J, ζ) . Root lattices, weight lattices:

Q

R

:=

X

r∈R

I

r

and Q

R^∨

:=

X

r∈R

J

r

P

R

:= {x ∈ V | ∀y ∈ Q

R^∨

, hx, yi ∈

Zk

} and P

R^∨

:= ... There is a Aut(R)/G(R)-invariant natural pairing

(P

R

/Q

R

) × (P

R^∨

/Q

R^∨

) → k/

Zk

.

(55)

Genera, Root and Weight Lattices

GL(V ) acts on root systems :

g · (I , J, ζ ) := (g (I ), g

^∨

(J), ζ ) . If

a

is a fractional ideal in k, we set

a

· (I, J, ζ) := (aI,

a^−∗

J, ζ) . Root lattices, weight lattices:

Q

R

:=

X

r∈R

I

r

and Q

R^∨

:=

X

r∈R

J

r

P

R

:= {x ∈ V | ∀y ∈ Q

R^∨

, hx, y i ∈

Zk

} and P

R^∨

:= ...

There is a Aut(R)/G(R)-invariant natural pairing

(P

R

/Q

R

) × (P

R^∨

/Q

R^∨

) → k/

Zk

.

(56)

Genera, Root and Weight Lattices

GL(V ) acts on root systems :

g · (I , J, ζ ) := (g (I ), g

^∨

(J), ζ ) . If

a

is a fractional ideal in k, we set

a

· (I, J, ζ) := (aI,

a^−∗

J, ζ) . Root lattices, weight lattices:

Q

R

:=

X

r∈R

I

r

and Q

R^∨

:=

X

r∈R

J

r

P

R

:= {x ∈ V | ∀y ∈ Q

R^∨

, hx, y i ∈

Zk

} and P

R^∨

:= ...

There is a Aut(R)/G(R)-invariant natural pairing (P

R

/Q

R

) × (P

R^∨

/Q

R^∨

) → k/

Zk

.

(57)

Classifying root systems

A root system is reduced if the map

r

7→ s

r

is injective. .

A root system is complete if the map

r

7→ s

_r

is surjective.

A root system is distinguished if the map

r

7→ s

_r

induces a bijection with the set of distinguished reflections.

Note that a distinguished root system is reduced, but that it is complete if and only if all reflections have order 2.

For each irreducible reflection group G , we provide a classification (up to genera),

over its ring of definition

Zk

,

of reduced complete root systems for G (de, e, r ),

of distinguished root systems corresponding to the 34 exceptional

groups.

(58)

Classifying root systems

A root system is reduced if the map

r

7→ s

r

is injective. . A root system is complete if the map

r

7→ s

_r

is surjective.

A root system is distinguished if the map

r

7→ s

_r

induces a bijection with the set of distinguished reflections.

Note that a distinguished root system is reduced, but that it is complete if and only if all reflections have order 2.

For each irreducible reflection group G , we provide a classification (up to genera),

over its ring of definition

Zk

,

of reduced complete root systems for G (de, e, r ),

of distinguished root systems corresponding to the 34 exceptional groups.

(59)

Classifying root systems

A root system is reduced if the map

r

7→ s

r

is injective. . A root system is complete if the map

r

7→ s

_r

is surjective.

A root system is distinguished if the map

r

7→ s

_r

induces a bijection with the set of distinguished reflections.

Note that a distinguished root system is reduced, but that it is complete if and only if all reflections have order 2.

For each irreducible reflection group G , we provide a classification (up to genera),

over its ring of definition

Zk

,

of reduced complete root systems for G (de, e, r ),

of distinguished root systems corresponding to the 34 exceptional

groups.

(60)

Classifying root systems

A root system is reduced if the map

r

7→ s

r

is injective. . A root system is complete if the map

r

7→ s

_r

is surjective.

A root system is distinguished if the map

r

7→ s

_r

induces a bijection with the set of distinguished reflections.

Note that a distinguished root system is reduced, but that it is complete if and only if all reflections have order 2.

For each irreducible reflection group G , we provide a classification (up to genera),

over its ring of definition

Zk

,

of reduced complete root systems for G (de, e, r ),

of distinguished root systems corresponding to the 34 exceptional groups.

(61)

Classifying root systems

A root system is reduced if the map

r

7→ s

r

is injective. . A root system is complete if the map

r

7→ s

_r

is surjective.

A root system is distinguished if the map

r

7→ s

_r

induces a bijection with the set of distinguished reflections.

Note that a distinguished root system is reduced, but that it is complete if and only if all reflections have order 2.

For each irreducible reflection group G , we provide a classification (up to genera),

over its ring of definition

Zk

,

of reduced complete root systems for G (de, e, r ),

of distinguished root systems corresponding to the 34 exceptional

groups.

(62)

Classifying root systems

A root system is reduced if the map

r

7→ s

r

is injective. . A root system is complete if the map

r

7→ s

_r

is surjective.

A root system is distinguished if the map

r

7→ s

_r

induces a bijection with the set of distinguished reflections.

Note that a distinguished root system is reduced, but that it is complete if and only if all reflections have order 2.

For each irreducible reflection group G , we provide a classification (up to genera),

over its ring of definition

Zk

, of reduced complete root systems for G (de, e, r ),

of distinguished root systems corresponding to the 34 exceptional groups.

(63)

Classifying root systems

A root system is reduced if the map

r

7→ s

r

is injective. . A root system is complete if the map

r

7→ s

_r

is surjective.

A root system is distinguished if the map

r

7→ s

_r