Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems and Coxeter groups
Root systems I
Root systems
Simple
systems
Hau-wen Huang
Department of Applied Mathematics, National Chiao Tung University, Taiwan
August 12, 2009
1 / 108
Content
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
I
Finite reflection groups
Root systems
Simple
systems
2 / 108
Content
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
I
Finite reflection groups
Root systems
I
Root systems
Simple
systems
3 / 108
Content
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
I
Finite reflection groups
Root systems
I
Root systems
Simple
systems
I
Simple systems
4 / 108
Content
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
I
Finite reflection groups
Root systems
I
Root systems
Simple
systems
I
Simple systems
I
Coxeter graphs
5 / 108
Content
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
I
Finite reflection groups
Root systems
I
Root systems
Simple
systems
I
Simple systems
I
Coxeter graphs
I
Classification of root systems (finite reflection groups)
6 / 108
References
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
J. E. Humphreys,
Reflection Groups and Coxeter Groups,
Cambridge University Press, Cambridge, 1990.
7 / 108
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Finite reflection groups
8 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
We are concerned with a fixed euclidean space E ,
Simple
systems
9 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
We are concerned with a fixed euclidean space E ,
i.e., a finite dimensional vector spacer over R endowed with a
positive definite symmetric bilinear form (λ, µ).
10 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
A reflection is a linear operator s on E which sends some
nonzero vector α to its negative while fixing pointwise the
hyperplane Hα orthogonal to α.
11 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
A reflection is a linear operator s on E which sends some
nonzero vector α to its negative while fixing pointwise the
hyperplane Hα orthogonal to α.
We may write s = sα .
12 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
13 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
r0
14 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
r0
?α
15 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
r
Hα
?α
16 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
sα α = −α
6
r
Hα
?α
17 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
r
@
@
Rλ
@
Hα
18 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
sα λ
r
@
@
Rλ
@
Hα
19 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Geometrically, sα is as follows.
µ
@
I
@
@r
Hα
20 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Geometrically, sα is as follows.
µ
@
I
@
@r
Hα
sα µ
21 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
r
-β
Hα
22 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Geometrically, sα is as follows.
Root systems
Simple
systems
r
-sα β = β
Hα
23 / 108
Finite reflection groups
Root systems
and Coxeter
groups
Basic properties for sα :
NCTS
Finite
reflection
groups
Root systems
Simple
systems
24 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Basic properties for sα :
1. sα = scα for any nonzero c ∈ R.
Finite
reflection
groups
Root systems
Simple
systems
25 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Basic properties for sα :
1. sα = scα for any nonzero c ∈ R.
2. sα2 = 1, in particular sα is invertible.
Root systems
Simple
systems
26 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Basic properties for sα :
1. sα = scα for any nonzero c ∈ R.
2. sα2 = 1, in particular sα is invertible.
3. sα is an orthogonal transformation, i.e.
(sα λ, sα µ) = (λ, µ) for all λ, µ ∈ E .
27 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Basic properties for sα :
1. sα = scα for any nonzero c ∈ R.
2. sα2 = 1, in particular sα is invertible.
3. sα is an orthogonal transformation, i.e.
(sα λ, sα µ) = (λ, µ) for all λ, µ ∈ E .
4. sα is in the orthogonal group O(E ) of E , consisting of all
orthogonal transformations of E .
28 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Basic properties for sα :
1. sα = scα for any nonzero c ∈ R.
2. sα2 = 1, in particular sα is invertible.
3. sα is an orthogonal transformation, i.e.
(sα λ, sα µ) = (λ, µ) for all λ, µ ∈ E .
4. sα is in the orthogonal group O(E ) of E , consisting of all
orthogonal transformations of E .
4. A simple formula:
sα λ = λ −
2(λ, α)
α.
(α, α)
29 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Definition
A finite reflection group is a finite group generated by
reflections.
30 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Definition
A finite reflection group is a finite group generated by
reflections.
Finite reflection groups are a type of finite subgroups of O(E ).
31 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Evidently, a finite reflection group W is generated by finite
reflections.
Simple
systems
32 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Evidently, a finite reflection group W is generated by finite
reflections.
Question
Is the converse true, i.e., is a group generated by finite
reflections finite?
33 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Root systems
Simple
systems
34 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Root systems
Simple
systems
r0
35 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Root systems
Simple
systems
β
6
r0
@
@
Rα
36 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Hα
Root systems
Simple
systems
β
6
r
@
@
Rα
37 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Hα
Root systems
Simple
systems
β
6
r
@
@
Rα
Hβ
38 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Hα
Root systems
Simple
systems
β
I
6
r
@
@
Rα
?
Hβ
39 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Hα
Root systems
Simple
systems
β
I
θ
6
r
@
@
Rα
?
Hβ
40 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Hα
Root systems
Simple
systems
β
I
θ
6
r
@
@
Rα
?
Hβ : x-axis
41 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Hα
Root systems
Simple
systems
β = (0, 1)
I
θ
6
r
@
@
Rα
?
Hβ : x-axis
42 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Relation between two reflections in a euclidean plane:
Hα
Root systems
Simple
systems
β = (0, 1)
I
θ
6
r
?
Hβ : x-axis
@
@
R α = (sin θ, − cos θ)
43 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Then
Finite
reflection
groups
Root systems
Simple
systems
r := sα sβ
cos 2θ
sin 2θ
1 0
=
sin 2θ − cos 2θ
0 −1
44 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Then
Finite
reflection
groups
Root systems
Simple
systems
r := sα sβ
cos 2θ
sin 2θ
1 0
=
sin 2θ − cos 2θ
0 −1
cos 2θ − sin 2θ
=
sin 2θ cos 2θ
45 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Then
Finite
reflection
groups
Root systems
Simple
systems
r := sα sβ
cos 2θ
sin 2θ
1 0
=
sin 2θ − cos 2θ
0 −1
cos 2θ − sin 2θ
=
sin 2θ cos 2θ
is a (counterclockwise) rotation through 2θ.
46 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
√
If θ = 42 π, the order of r is infinite. Thus, the answer of that
question is negative.
Root systems
Simple
systems
47 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
√
If θ = 42 π, the order of r is infinite. Thus, the answer of that
question is negative.
Example I:
Root systems
Simple
systems
48 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
√
If θ = 42 π, the order of r is infinite. Thus, the answer of that
question is negative.
Example I:
k
Assume that θ = m
π where k and m are two coprime positive
m
integers, then r = 1 and the group Dm generated by {sα , sβ }
consists of
49 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
√
If θ = 42 π, the order of r is infinite. Thus, the answer of that
question is negative.
Example I:
k
Assume that θ = m
π where k and m are two coprime positive
m
integers, then r = 1 and the group Dm generated by {sα , sβ }
consists of
{1, r , . . . , r m−1 , sα , rsα , . . . , r m−1 sα }.
50 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
√
If θ = 42 π, the order of r is infinite. Thus, the answer of that
question is negative.
Example I:
k
Assume that θ = m
π where k and m are two coprime positive
m
integers, then r = 1 and the group Dm generated by {sα , sβ }
consists of
{1, r , . . . , r m−1 , sα , rsα , . . . , r m−1 sα }.
Note that {1, r , . . . , r m−1 } consists of m rotations (through
multiples of 2π
m ).
51 / 108
Finite reflection groups
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Our goal is to classify all finite reflection groups.
52 / 108
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Root systems
53 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
We always assume that W is a finite reflection group.
Root systems
Simple
systems
54 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
We always assume that W is a finite reflection group.
Proposition
If t ∈ O(E ) and µ is any nonzero vector in E , then
tsµ t −1 = stµ .
55 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
We always assume that W is a finite reflection group.
Proposition
If t ∈ O(E ) and µ is any nonzero vector in E , then
tsµ t −1 = stµ . In particular, if ω ∈ W , then sωµ ∈ W whenever
sµ ∈ W .
Proof.
56 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
We always assume that W is a finite reflection group.
Proposition
If t ∈ O(E ) and µ is any nonzero vector in E , then
tsµ t −1 = stµ . In particular, if ω ∈ W , then sωµ ∈ W whenever
sµ ∈ W .
Proof. Easy verification.
57 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
A finite set Φ of nonzero (unit) vectors in E is called a root
system in E if the following axioms are satisfied:
Simple
systems
58 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
A finite set Φ of nonzero (unit) vectors in E is called a root
system in E if the following axioms are satisfied:
(R1) Φ ∩ Rµ = {µ, −µ} for all µ ∈ Φ;
59 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
A finite set Φ of nonzero (unit) vectors in E is called a root
system in E if the following axioms are satisfied:
(R1) Φ ∩ Rµ = {µ, −µ} for all µ ∈ Φ;
(R2) sµ Φ = Φ for all µ ∈ Φ.
60 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
A finite set Φ of nonzero (unit) vectors in E is called a root
system in E if the following axioms are satisfied:
(R1) Φ ∩ Rµ = {µ, −µ} for all µ ∈ Φ;
(R2) sµ Φ = Φ for all µ ∈ Φ.
The elements of Φ are called roots.
61 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Example II:
Let the notations as Example I. Then the set Ψ = Dm {α, β} is
a root system and sµ (µ ∈ Ψ) generate Dm .
Root systems
Simple
systems
62 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Example II:
Let the notations as Example I. Then the set Ψ = Dm {α, β} is
a root system and sµ (µ ∈ Ψ) generate Dm . For example, if
m = 4, then Ψ is as follows.
Root systems
Simple
systems
63 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Example II:
Let the notations as Example I. Then the set Ψ = Dm {α, β} is
a root system and sµ (µ ∈ Ψ) generate Dm . For example, if
m = 4, then Ψ is as follows.
Root systems
Simple
systems
r0
64 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Example II:
Let the notations as Example I. Then the set Ψ = Dm {α, β} is
a root system and sµ (µ ∈ Ψ) generate Dm . For example, if
m = 4, then Ψ is as follows.
Root systems
6
Simple
systems
@
I
@
@
@
@r
@
@
@
@
@
R
?
65 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Example II:
Let the notations as Example I. Then the set Ψ = Dm {α, β} is
a root system and sµ (µ ∈ Ψ) generate Dm . For example, if
m = 4, then Ψ is as follows.
Root systems
6
Simple
systems
@
I
I
@
@
@
@r
@
@
π π
4 (m)
?
@
@
@
R
?
66 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
It is immediate from the above proposition that every finite
reflection group W is generated by {sµ | µ ∈ Φ} for some root
system Φ.
Root systems
Simple
systems
67 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
It is immediate from the above proposition that every finite
reflection group W is generated by {sµ | µ ∈ Φ} for some root
system Φ.
Conversely, any group W arising from a root system is finite:
Simple
systems
68 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
It is immediate from the above proposition that every finite
reflection group W is generated by {sµ | µ ∈ Φ} for some root
system Φ.
Conversely, any group W arising from a root system is finite:
Observe that there is a homomorphism f of W into the
symmetric group on Φ, f (ω)µ := ωµ for ω ∈ W and µ ∈ Φ.
69 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
It is immediate from the above proposition that every finite
reflection group W is generated by {sµ | µ ∈ Φ} for some root
system Φ.
Conversely, any group W arising from a root system is finite:
Observe that there is a homomorphism f of W into the
symmetric group on Φ, f (ω)µ := ωµ for ω ∈ W and µ ∈ Φ.
Since each sµ (µ ∈ Φ) and hence each element of W fixes
pointwise the orthogonal complement of the subspace spanned
by Φ, only ω = 1 can fix all elements of Φ.
70 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
It is immediate from the above proposition that every finite
reflection group W is generated by {sµ | µ ∈ Φ} for some root
system Φ.
Conversely, any group W arising from a root system is finite:
Observe that there is a homomorphism f of W into the
symmetric group on Φ, f (ω)µ := ωµ for ω ∈ W and µ ∈ Φ.
Since each sµ (µ ∈ Φ) and hence each element of W fixes
pointwise the orthogonal complement of the subspace spanned
by Φ, only ω = 1 can fix all elements of Φ. This means that
the kernel of f is trivial, and forces W to be finite.
71 / 108
Root systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
It is immediate from the above proposition that every finite
reflection group W is generated by {sµ | µ ∈ Φ} for some root
system Φ.
Conversely, any group W arising from a root system is finite:
Observe that there is a homomorphism f of W into the
symmetric group on Φ, f (ω)µ := ωµ for ω ∈ W and µ ∈ Φ.
Since each sµ (µ ∈ Φ) and hence each element of W fixes
pointwise the orthogonal complement of the subspace spanned
by Φ, only ω = 1 can fix all elements of Φ. This means that
the kernel of f is trivial, and forces W to be finite.
72 / 108
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Simple systems
73 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
We always assume that Φ is a root system in E and W is the
finite reflection group generated by sµ , µ ∈ Φ.
74 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
A subset ∆ of Φ is called a simple system if the following
conditions are satisfied:
Root systems
Simple
systems
75 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
A subset ∆ of Φ is called a simple system if the following
conditions are satisfied:
(i) ∆ is a vector space basis for the R-span of Φ in E ;
Simple
systems
76 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
A subset ∆ of Φ is called a simple system if the following
conditions are satisfied:
(i) ∆ is a vector space basis for the R-span of Φ in E ;
(ii) each root in Φ is a linear combination of ∆ with
coefficients all of the same sign.
77 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
A subset ∆ of Φ is called a simple system if the following
conditions are satisfied:
(i) ∆ is a vector space basis for the R-span of Φ in E ;
(ii) each root in Φ is a linear combination of ∆ with
coefficients all of the same sign.
The proof of the existence of simple systems is omitted.
78 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
A subset ∆ of Φ is called a simple system if the following
conditions are satisfied:
(i) ∆ is a vector space basis for the R-span of Φ in E ;
(ii) each root in Φ is a linear combination of ∆ with
coefficients all of the same sign.
The proof of the existence of simple systems is omitted.
All roots in ∆ are call simple roots.
79 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Example III:
Let the notations as Example II. Determine all simple systems
in Ψ. For example, if m = 4, then Ψ is as follows.
Root systems
Simple
systems
80 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Example III:
Let the notations as Example II. Determine all simple systems
in Ψ. For example, if m = 4, then Ψ is as follows.
Root systems
6
Simple
systems
@
I
I
@
@
@
@r
@
@
π π
4 (m)
?
@
@
@
R
?
81 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Example III:
Let the notations as Example II. Determine all simple systems
in Ψ. For example, if m = 4, then Ψ is as follows.
Root systems
6
Simple
systems
@
I
I
@
@
@
@r
@
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82 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Fix a simple system ∆ in Φ.
Root systems
Simple
systems
83 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Fix a simple system ∆ in Φ. Let Π denote the subset of Φ
consisting of all roots which are nonnegative linear
combinations of ∆.
84 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Fix a simple system ∆ in Φ. Let Π denote the subset of Φ
consisting of all roots which are nonnegative linear
combinations of ∆. In particular, ∆ ⊂ Π.
85 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Fix a simple system ∆ in Φ. Let Π denote the subset of Φ
consisting of all roots which are nonnegative linear
combinations of ∆. In particular, ∆ ⊂ Π. The set Π is called
the positive system of ∆, and all roots in Π are called
positive roots. (relative to ∆).
86 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Basic properties for ∆ :
Root systems
Simple
systems
87 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Basic properties for ∆ :
1. For µ ∈ Φ, precisely one of each pair {µ, −µ} is in Π.
Root systems
Simple
systems
88 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Basic properties for ∆ :
1. For µ ∈ Φ, precisely one of each pair {µ, −µ} is in Π.
Root systems
Simple
systems
2. All positive (resp. simple) systems have the same
cardinality.
89 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Basic properties for ∆ :
1. For µ ∈ Φ, precisely one of each pair {µ, −µ} is in Π.
Root systems
Simple
systems
2. All positive (resp. simple) systems have the same
cardinality.
3. ω∆ is a simple system for ω ∈ W .
90 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Basic properties for ∆ :
1. For µ ∈ Φ, precisely one of each pair {µ, −µ} is in Π.
Root systems
Simple
systems
2. All positive (resp. simple) systems have the same
cardinality.
3. ω∆ is a simple system for ω ∈ W .
4. The positive system of ω∆ is ωΠ.
91 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Consider the special case ω = sδ (δ ∈ ∆). We find that Π and
sδ Π differ only by one root:
Root systems
Simple
systems
92 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Consider the special case ω = sδ (δ ∈ ∆). We find that Π and
sδ Π differ only by one root:
Root systems
Proposition
Simple
systems
For δ ∈ ∆, we have sδ (Π \ {δ}) = Π \ {δ}
Proof.
93 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Consider the special case ω = sδ (δ ∈ ∆). We find that Π and
sδ Π differ only by one root:
Root systems
Proposition
Simple
systems
For δ ∈ ∆, we have sδ (Π \ {δ}) = Π \ {δ}
P
Proof. Let µ ∈ Π, µ 6= δ. Then µ = δ0 ∈∆ cδ0 δ 0 for some
cδ0 > 0 with δ 0 6= δ.
94 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Consider the special case ω = sδ (δ ∈ ∆). We find that Π and
sδ Π differ only by one root:
Root systems
Proposition
Simple
systems
For δ ∈ ∆, we have sδ (Π \ {δ}) = Π \ {δ}
P
Proof. Let µ ∈ Π, µ 6= δ. Then µ = δ0 ∈∆ cδ0 δ 0 for some
cδ0 > 0 with δ 0 6= δ. Since sδ µ is obtained from µ by subtracting
a multiple of δ, the root sδ µ is still positive and it is not δ. 95 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Theorem
W is generated by the reflections sδ (δ ∈ ∆)
Root systems
Simple
systems
Proof.
96 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Theorem
W is generated by the reflections sδ (δ ∈ ∆)
Root systems
Simple
systems
Proof. Let W 0 denote the subgroup of W generated by sδ ,
δ ∈ ∆. By the above proposition, it suffices to show that
Π ⊂ W 0 ∆.
97 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Theorem
W is generated by the reflections sδ (δ ∈ ∆)
Root systems
Simple
systems
Proof. Let W 0 denote the subgroup of W generated by sδ ,
δ ∈ ∆. By the above proposition, it suffices toPshow that
Π ⊂ W 0 ∆. For
P µ ∈ Φ, let the height of µ = δ∈∆ cδ δ be the
real number δ∈∆ cδ , abbreviated ht(µ). Suppose γP
∈ Π is the
smallest height root that is not in W 0 ∆. Write γ = δ∈∆ cδ δ.
98 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Note that 0 < (γ, γ) =
some δ ∈ ∆.
P
δ∈∆ cδ (γ, δ),
forcing (γ, δ) > 0 for
Root systems
Simple
systems
99 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
P
Note that 0 < (γ, γ) = δ∈∆ cδ (γ, δ), forcing (γ, δ) > 0 for
some δ ∈ ∆. Since sδ γ is obtained from γ by subtracting a
positive multiple of δ, we have ht(sδ γ) < ht(γ).
Simple
systems
100 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
P
Note that 0 < (γ, γ) = δ∈∆ cδ (γ, δ), forcing (γ, δ) > 0 for
some δ ∈ ∆. Since sδ γ is obtained from γ by subtracting a
positive multiple of δ, we have ht(sδ γ) < ht(γ). But sδ γ is
positive (since γ 6= δ and by the above proposition) and hence
it is also not in W 0 ∆, contradicting the choice of γ.
101 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
P
Note that 0 < (γ, γ) = δ∈∆ cδ (γ, δ), forcing (γ, δ) > 0 for
some δ ∈ ∆. Since sδ γ is obtained from γ by subtracting a
positive multiple of δ, we have ht(sδ γ) < ht(γ). But sδ γ is
positive (since γ 6= δ and by the above proposition) and hence
it is also not in W 0 ∆, contradicting the choice of γ.
Corollary (of proof)
Φ = W ∆.
102 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Let ∆0 be another simple system in Φ, and let Π0 be the
positive system of the simple system ∆0 . Observe that if
Π = Π0 then ∆ = ∆0 (Exercise).
103 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Root systems
Theorem
Let ∆ and ∆0 be two simple systems in Φ. Then ∆0 = ω∆ for
some ω ∈ W .
Simple
systems
Proof.
Finite
reflection
groups
104 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Theorem
Let ∆ and ∆0 be two simple systems in Φ. Then ∆0 = ω∆ for
some ω ∈ W .
Proof. Let Π and Π0 be the two positive systems of ∆ and ∆0
respectively. It is enough to prove Π0 = ωΠ for some ω ∈ W .
105 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Theorem
Let ∆ and ∆0 be two simple systems in Φ. Then ∆0 = ω∆ for
some ω ∈ W .
Proof. Let Π and Π0 be the two positive systems of ∆ and ∆0
respectively. It is enough to prove Π0 = ωΠ for some ω ∈ W . If
Card(Π ∩ Π0 ) = Card(Π) i.e. Π = Π0 , then ω = 1 are satisfied.
106 / 108
Simple systems
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Theorem
Let ∆ and ∆0 be two simple systems in Φ. Then ∆0 = ω∆ for
some ω ∈ W .
Proof. Let Π and Π0 be the two positive systems of ∆ and ∆0
respectively. It is enough to prove Π0 = ωΠ for some ω ∈ W . If
Card(Π ∩ Π0 ) = Card(Π) i.e. Π = Π0 , then ω = 1 are satisfied.
Otherwise, choose δ ∈ ∆ with −δ ∈ Π0 . The above proposition
implies that Card(sδ Π ∩ Π0 ) = Card(Π ∩ Π0 ) + 1. Iterate!! 107 / 108
Root systems
and Coxeter
groups
NCTS
Finite
reflection
groups
Root systems
Simple
systems
Thanks for your attention.
108 / 108