Coxeter group

Coxeter group

A

Coxeter group

G

is a

finitely generated group

, which carries a

presentation

of the form

W

=

⟨

w

1

,

…

,

w

n

∣

(

w

i

⁢

w

j

)

m

i

⁢

j

=

1

⟩

where the integers

m

i

⁢

j

satisfy

m

i

⁢

i

=

1

for

i

=

1

,

…

,

n

and

m

i

⁢

j

=

m

j

⁢

i

≥

2

for

i

≠

j

. The

exponents

form a matrix

[

m

i

⁢

j

]

1

≤

i

,

j

≤

n

often called the

Coxeter matrix

.
This is a cousin of the

Cartan matrix

and both encode the information
of the

Dynkin diagrams

.

A Dynkin diagram is the graph with the

adjacency matrix

given by

[

m

i

⁢

j

-

2

]

1

≤

i

,

j

≤

n

where

[

m

i

⁢

j

]

1

≤

i

,

j

≤

n

is a
Coxeter matrix.

A finite Coxeter group is

irreducible

if it is not the

direct product

of smaller coxeter groups. These groups are classified and labeled
labeled by the Bourbaki types

𝖠

n

,

𝖡

n

,

𝖢

n

,

𝖣

n

,

𝖤

6

,

𝖤

7

,

𝖤

8

,

𝖥

4

,

𝖦

2

,

𝖧

2

n

,

𝖨

3

,

𝖨

4

.

The

classification

depends on realizing the groups as

reflections

of

hyperplanes

in a finite dimensional

real vector space

.
Then observing a condition on an

inner product

to be integer valued, it
is possible to show these families of

symmetry

are all that can exist.
The Cartan matrix encodes these integer values of the inner product
of

adjacent

reflections while the Coxeter matrix encodes the orders of
adjacent

products

of

generators

.

Remark 1

.

The notation

A

n

should not be confused with the natation for
the

alternating group

on

n

elements,

A

n

. This unfortunate overlap is
also a problem with

D

n

which is not the same as the

dihedral
group

on

n

-vertices,

D

n

.

Alternative methods to study Coxeter groups is through the use of a length
measurement on elements in the group. As every element in

g

in a Coxeter
group is the product of the

involutions

w

1

,

…

,

w

n

, the

length

is defined as the shortest word in these

w

i

′

⁢

s

to equal

g

. We denote this

ł

⁢

(

g

)

. Then using careful
analysis and the

exchange condition

it is also possible to specify many of the necessary

properties

of irreducible Coxeter groups.

Recall that a

Weyl group

W

is a group generated by involutions

S

, that is,
generated by elements of order 2. The

exchange condition

on a

W

with respect to

S

states that given a

reduced word

w

=

w

i

1

⁢

⋯

⁢

w

i

k

in

W

,

w

i

∈

S

, such that for every

s

∈

S

,

ł

⁢

(

s

⁢

w

)

≤

ł

⁢

(

w

)

then there exists an

j

such that

s

⁢

w

=

w

i

1

⁢

⋯

⁢

w

i

j

-

1

⁢

w

i

j

+

1

⁢

⋯

⁢

w

i

k

.

The insistence that

w

i

2

=

1

shows that Coxeter groups are generated by
involutions. This makes every Coxeter group a Weyl group. However,
not every Weyl group is a Coxeter group.

The remaining condition to make a Weyl group a Coxeter group is the exchange condition. Thus every finite Weyl group with the exchange condition is a
Coxeter group, and visa-versa.

Coxeter groups arrise as the Weyl groups of

Lie algebra

, Lie groups, and groups of with a

BN-pair

. However many other usese exist. It should be noted
that the study of Lie theory makes use only of the

crystallographic

coxeter groups, which are those of type

𝖠

n

,

𝖡

n

,

𝖢

n

,

𝖣

n

,

𝖤

6

,

𝖤

7

,

𝖤

8

,

𝖥

4

,

𝖦

2

.

Thus it omits

𝖧

2

n

,

𝖨

3

and

𝖨

4

1

Coxeter groups as reflections

Let us see more concretely how a finite Coxeter group can be realized.

Let

V

be a real

Euclidean vector space

and

𝒪

⁢

(

V

)

the group of all

orthogonal transformations

of

V

.

A

reflection

of

V

is a

linear transformation

S

that carries each vector to its

mirror image

with respect to a fixed hyperplane

𝒫

; it is clear geometrically that a reflection is an orthogonal transformation.

A

subgroup

𝒢

≤

𝒪

⁢

(

V

)

will be called

effective

if

V

0

⁢

(

𝒢

)

=

0

where

V

0

⁢

(

𝒢

)

=

⋂

T

∈

𝒢

{

x

∈

V

∣

T

⁢

x

=

x

}

.

A finite Coxeter group can be realized as (i.e. is always

isomorphic

to) a finite effective subgroup

𝒢

of

𝒪

⁢

(

V

)

that is generated by a set of reflections, for some

Euclidean space

V

.

2

Classification of irreducible finite Coxeter groups

Type

𝖠

n

: This group is isomorphic to the

symmetric group

on

n

elements,

S

n

. The coxeter matrix is encoded by

m

i

,

i

+

1

=

3

=

m

i

+

1

,

i

and all other terms are 2. To observe the

isomorphism

let

w

1

=

(

1

,

2

)

,

(

2

,

3

)

,

…

,

w

n

=

(

n

-

1

,

n

)

.

Then

w

i

2

=

1

, for instance

(

1

,

2

)

2

=

(

)

,

(

w

i

⁢

w

j

)

2

=

1

if

|

i

-

j

|

>

1

, for example

(

(

1

,

2

)

⁢

(

3

,

4

)

)

2

=

0

and

(

w

i

⁢

w

i

+

1

)

3

=

1

as we see with

(

1

,

2

)

⁢

(

2

,

3

)

=

(

1

,

2

,

3

)

which has order 3.

The Dynkin diagram is:

\xymatrix

∘

\ar

@

-

[

r

]

&

∘

\ar

@

-

[

r

]

&

∘

\ar

@

-

-

[

r

]

&

∘

.

Type

𝖡

n

,

𝖢

n

: This group is isomorphic to the

wreath product

ℤ

2

≀

S

n

, that is, the

semi-direct product

of

ℤ

2

n

⋊

S

n

where

S

n

permutes the entries of the vectors in

ℤ

2

n

.

The designation of type

𝖡

n

and

𝖢

n

relate to the fact that two different methods can be given to

construct

the same group (as the Weyl group of

O

⁢

(

2

⁢

n

+

1

,

k

)

or as the Weyl group of

S

⁢

p

⁢

(

2

⁢

n

,

k

)

).
It is also common to see

𝖢

n

used as the sole label.

Type

𝖧

2

n

: These groups are the dihedral group

D

2

⁢

n

for

n

≥

5

and

n

≠

6

.

Type

𝖦

2

: This group is isomorphic to

S

3

.

References

L. C. Grove, C. T. Benson,

Finite Reflection Groups. Second Edition.

, Springer-Verlag, 1985.

Title

Coxeter group

Canonical

name

CoxeterGroup

Date of creation

2013-03-22 15:38:20

Last modified on

2013-03-22 15:38:20

Owner

Simone (5904)

Last modified by

Simone (5904)

Numerical id

12

Author

Simone (5904)

Entry type

Definition

Classification

msc 20F55

Defines

Coxeter group

Defines

Coxeter matrix

Defines

Cartan matrix

Defines

Weyl group

Defines

exchange condition

Defines

length of Weyl group

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