Root systems and Coxeter groups - National Chiao Tung ...weng/courses/2009...and Coxeter groups NCTS Radical of the bilinear form Dual representation Finite Coxeter groups References

Root systemsand Coxeter

groups

NCTS

Radical of thebilinear form

Dualrepresentation

Finite Coxetergroups

Root systems and Coxeter groupsCoxeter groups II

Hau-wen Huang

Department of Applied Mathematics, National Chiao Tung University, Taiwan

August 13, 2009

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Content

I Coxeter groups

I Length function

I Geometric representation of W

I Geometric interpretation of the length function

I Radical of the bilinear form

I Dual representation

I Finite Coxeter groups

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Dualrepresentation


Content

I Coxeter groups

I Length function






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NCTS


Dualrepresentation


Content

I Coxeter groups

I Length function






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NCTS


Dualrepresentation


Content

I Coxeter groups

I Length function






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Dualrepresentation


Content

I Coxeter groups

I Length function






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Dualrepresentation


Content

I Coxeter groups

I Length function






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Dualrepresentation


Content

I Coxeter groups

I Length function






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References

N. Bourbaki,Lie Groups and Lie algebras: Chapters 4-6,Springer-Verlag, Berlin, 2002.

J. E. Humphreys,Reflection Groups and Coxeter Groups,Cambridge University Press, Cambridge, 1990.

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Radical of the bilinear form

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Our task of this section is to show:

If W is infinite, then the center Z (W ) of W is trivial.

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Recall that

I the definition of the irreducible Coxeter system (W ,S).

I the bilinear form B defined by

B(αs , αs′) := − cosπ

m(s, s ′).

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The radical of B is

V⊥ := {λ ∈ V | B(λ, µ) = 0 ∀µ ∈ V }.

We say that B on V is nondegenerate if V⊥ = {0};otherwise, B is degenerate.

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Basic properties for V⊥ :

I V⊥ is a W -invariant proper subspace.

I V⊥ =⋂

s∈S Hs .

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I V⊥ =⋂

s∈S Hs .

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I V⊥ =⋂

s∈S Hs .

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Proposition

Assume that (W ,S) is irreducible. Every proper W -invariantsubspace of V is included in the radical V⊥ of the form B.

Proof.

Let V ′ be a W -invariant proper subspace. Irreducibilityfor (W ,S) implies that no root αs (s ∈ S) lies in V ′. For anyλ ∈ V ′ and s ∈ S , since σsλ− λ ∈′ V , we have B(αs , λ) = 0.Hence V ′ lies in

⋂s∈S Hs = V

⊥. �

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Proposition


Proof. Let V ′ be a W -invariant proper subspace. Irreducibilityfor (W ,S) implies that no root αs (s ∈ S) lies in V ′.

For anyλ ∈ V ′ and s ∈ S , since σsλ− λ ∈′ V , we have B(αs , λ) = 0.Hence V ′ lies in

⋂s∈S Hs = V

⊥. �

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Proposition


Proof. Let V ′ be a W -invariant proper subspace. Irreducibilityfor (W ,S) implies that no root αs (s ∈ S) lies in V ′. For anyλ ∈ V ′ and s ∈ S , since σsλ− λ ∈′ V , we have B(αs , λ) = 0.

Hence V ′ lies in⋂

s∈S Hs = V⊥. �

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USER註解move ' to the upper right corner of V


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Proposition


Proof. Let V ′ be a W -invariant proper subspace. Irreducibilityfor (W ,S) implies that no root αs (s ∈ S) lies in V ′. For anyλ ∈ V ′ and s ∈ S , since σsλ− λ ∈′ V , we have B(αs , λ) = 0.Hence V ′ lies in

⋂s∈S Hs = V

⊥. �

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Corollary

Assume that (W ,S) is irreducible.

(i) If B is degenerate, then V fails to be completely reducibleas a W -module.

(ii) If B is nondegenerate, then V is irreducible as aW -module.

�

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Corollary

Assume that (W ,S) is irreducible. The only endomorphisms ofV commuting with the action of W are the scalars.

Proof.

Let z be an endomorphism of V commuting with all σsfor s ∈ S . Fix any s ∈ S . From zσs = σsz acting on αs , wefind zαs = cαs for some scalar c . Claim that z = c · 1. Observethat kernel V ′ of z − c · 1 is a W -invariant subspace andV ′ 6= {0}. Thanks to the above proposition and αs ∈ V ′, wemust have V ′ = V . �

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Corollary


Proof. Let z be an endomorphism of V commuting with all σsfor s ∈ S . Fix any s ∈ S . From zσs = σsz acting on αs , wefind zαs = cαs for some scalar c .

Claim that z = c · 1. Observethat kernel V ′ of z − c · 1 is a W -invariant subspace andV ′ 6= {0}. Thanks to the above proposition and αs ∈ V ′, wemust have V ′ = V . �

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Corollary


Proof. Let z be an endomorphism of V commuting with all σsfor s ∈ S . Fix any s ∈ S . From zσs = σsz acting on αs , wefind zαs = cαs for some scalar c . Claim that z = c · 1.

Observethat kernel V ′ of z − c · 1 is a W -invariant subspace andV ′ 6= {0}. Thanks to the above proposition and αs ∈ V ′, wemust have V ′ = V . �

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Corollary


Proof. Let z be an endomorphism of V commuting with all σsfor s ∈ S . Fix any s ∈ S . From zσs = σsz acting on αs , wefind zαs = cαs for some scalar c . Claim that z = c · 1. Observethat kernel V ′ of z − c · 1 is a W -invariant subspace andV ′ 6= {0}. Thanks to the above proposition and αs ∈ V ′, wemust have V ′ = V . �

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Exercise


Proof.

Let z ∈ Z (W ). By the above corollary, z = c · 1 forsome scalar c . Since z preserves B, we have c = ±1. But−1 /∈ σ(W ) when W is infinite. �

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Exercise


Proof. Let z ∈ Z (W ). By the above corollary, z = c · 1 forsome scalar c .

Since z preserves B, we have c = ±1. But−1 /∈ σ(W ) when W is infinite. �

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Exercise


Proof. Let z ∈ Z (W ). By the above corollary, z = c · 1 forsome scalar c . Since z preserves B, we have c = ±1. But−1 /∈ σ(W ) when W is infinite. �

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Dual representation

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Dual representation

Let V ∗ be the dual space of V . The dual representation of σis σ∗ : W → GL(V ∗) defined by

σ∗(ω) := tσ(ω−1)

for ω ∈ W .

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Dual representation

For f ∈ V ∗, we write ω(f ) in place of σ∗(ω)(f ).

For s ∈ S , let

As := {f ∈ V ∗ | f (αs) > 0}.

Let C :=⋂

s∈S As .

Note that A′s := {f ∈ V ∗ | f (αs) < 0} = s(As).

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Dual representation

For f ∈ V ∗, we write ω(f ) in place of σ∗(ω)(f ).For s ∈ S , let

As := {f ∈ V ∗ | f (αs) > 0}.

Let C :=⋂

s∈S As .


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Dual representation

For f ∈ V ∗, we write ω(f ) in place of σ∗(ω)(f ).For s ∈ S , let

As := {f ∈ V ∗ | f (αs) > 0}.

Let C :=⋂

s∈S As .


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Dual representation

The dual version of geometric characterization for the lengthfunction ` :

Lemma

Let s ∈ S and ω ∈ W . Then `(sω) > `(ω) if and only ifω(C ) ⊂ As , whereas `(sω) < `(ω) if and only if ω(C ) ⊂ A′s .

Proof.

`(sω) > `(ω)⇔ `(ω−1s) > `(ω−1)⇔ ω−1(αs) > 0⇔ 0 < f (ω−1(αs)) = ω(f )(αs) ∀f ∈ C⇔ ω(C ) ⊂ As .

�

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Dual representation

The dual version of geometric characterization for the lengthfunction ` :

Lemma

Let s ∈ S and ω ∈ W . Then `(sω) > `(ω) if and only ifω(C ) ⊂ As , whereas `(sω) < `(ω) if and only if ω(C ) ⊂ A′s .

Proof.

`(sω) > `(ω)⇔ `(ω−1s) > `(ω−1)⇔ ω−1(αs) > 0⇔ 0 < f (ω−1(αs)) = ω(f )(αs) ∀f ∈ C⇔ ω(C ) ⊂ As .

�

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Dual representation

Theorem

(Tits). If ω ∈ W and C ∩ ω(C ) 6= ∅, then ω = 1.

Proof.

Suppose `(ω) > 0. Then there exists some s ∈ S forwhich `(sω) < `(ω). By the above lemma, we obtain thatω(C ) ⊂ A′s , which is contradiction to C ∩ ω(C ) 6= ∅. Thusω = 1. �

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Dual representation

Theorem


Proof. Suppose `(ω) > 0. Then there exists some s ∈ S forwhich `(sω) < `(ω).

By the above lemma, we obtain thatω(C ) ⊂ A′s , which is contradiction to C ∩ ω(C ) 6= ∅. Thusω = 1. �

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Dual representation

Theorem


Proof. Suppose `(ω) > 0. Then there exists some s ∈ S forwhich `(sω) < `(ω). By the above lemma, we obtain thatω(C ) ⊂ A′s , which is contradiction to C ∩ ω(C ) 6= ∅. Thusω = 1. �

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Dual representation

Let |S | = n. We identify V with Rn, say by fixing the basis{αs | s ∈ S}. Then V ∗ with the dual basis may be identifiedwith Rn, and GL(V ∗) with GL(n,R). Also, GL(n,R) can beviewed as a subspace of Rn

2.

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Dual representation

Consider the standard topological spaces Rn and Rn2. For any

fixed f ∈ V ∗, the orbit map GL(V ∗) → V ∗ sending g 7→ g · fis continuous (Exercise).

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Dual representation

Choose f ∈ C . Note that the inverse image U of C is an openneighborhood of 1 ∈ GL(V ∗).

By the above theorem,

σ∗(W ) ∩ U = {1}.

In turn, an element g ∈ σ∗(W ) has an open neighborhood gUintersecting σ∗(W ) in {g}. This means that σ∗(W ) is adiscrete subset of GL(V ∗).

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Dual representation

Choose f ∈ C . Note that the inverse image U of C is an openneighborhood of 1 ∈ GL(V ∗). By the above theorem,

σ∗(W ) ∩ U = {1}.


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σ∗(W ) ∩ U = {1}.

In turn, an element g ∈ σ∗(W ) has an open neighborhood gUintersecting σ∗(W ) in {g}.

This means that σ∗(W ) is adiscrete subset of GL(V ∗).

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Dual representation


σ∗(W ) ∩ U = {1}.


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Dual representation

By transport of structure, we obtain

Proposition

σ(W ) is a discrete subgroup of GL(V ). �

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Dual representation

Corollary

If the form B is positive definite, then W is finite.

Proof. Since B is positive definite, there is an orthonormalbasis of V .

We may identify σ(W ) with a subgroup of theorthogonal group O(n,R). It is well-known that O(n,R) is acompact subset of GL(n,R). Since a discrete subset of acompact space is finite, W ∼= σ(W ) is finite. �

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Dual representation

Corollary


Proof. Since B is positive definite, there is an orthonormalbasis of V . We may identify σ(W ) with a subgroup of theorthogonal group O(n,R). It is well-known that O(n,R) is acompact subset of GL(n,R).

Since a discrete subset of acompact space is finite, W ∼= σ(W ) is finite. �

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Dual representation

Corollary


Proof. Since B is positive definite, there is an orthonormalbasis of V . We may identify σ(W ) with a subgroup of theorthogonal group O(n,R). It is well-known that O(n,R) is acompact subset of GL(n,R). Since a discrete subset of acompact space is finite, W ∼= σ(W ) is finite. �

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Finite Coxeter groups

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Review some standard facts about group representations.

Lemma

Let ρ : G → GL(E ) be a group representation, with E a finitedimensional vector space over R.

(i) If G is finite, then there exists a positive definiteG-invariant bilinear form on E .

(ii) If G is finite, then ρ is completely reducible.

(iii) Suppose the only endomorphisms of E commuting withρ(G ) are the scalars. If β and β′ are nondegeneratesymmetric bilinear forms on E , both G-invariant, then β′

is a scalar multiple of β.

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Proof.

(i) Start with any positive definite symmetric bilinearform β on E . Then

α(λ, µ) :=∑g∈G

β(g · λ, g · µ)

as desired.

(ii) The orthogonal complement of a G -invariant subspace(relative to α) is also G -invariant, so complete reducibilityfollows.

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Proof. (i) Start with any positive definite symmetric bilinearform β on E . Then

α(λ, µ) :=∑g∈G

β(g · λ, g · µ)

as desired.


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Proof. (i) Start with any positive definite symmetric bilinearform β on E . Then

α(λ, µ) :=∑g∈G

β(g · λ, g · µ)

as desired.


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(iii) The following diagram is commutative for all g ∈ G .

-

-

-

E E

E ∗ E ∗

E E

? ?

? ?

ρ(g)

ρ∗(g)

ρ(g)

β β

β′−1 β′−1

By assumption, this is just a scalar, so β′ is a scalar multiple ofβ. �

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(iii) The following diagram is commutative for all g ∈ G .-

-

-

E E

E ∗ E ∗

E E

? ?

? ?

ρ(g)

ρ∗(g)

ρ(g)

β β

β′−1 β′−1


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-

-

-

-

E E

E ∗ E ∗

E E

? ?

? ?? ?

ρ(g)

ρ∗(g)

ρ(g)

β β

β′−1 β′−1


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-

-

-

-

E E

E ∗ E ∗

E E

? ?

? ?? ?

ρ(g)

ρ∗(g)

ρ(g)

β β

β′−1 β′−1


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Theorem

The following conditions on the irreducible Coxeter system(W ,S) are equivalent:

(i) W is finite.

(ii) The bilinear form B is positive definite.

Sketch of Proof.

(i) ⇒ (ii) Thanks to part (b) of the lemmaabove, W acts completely reducibly on V . Then B must benondegenerate, and the scalars are the only endomorphisms ofV commuting with the action of W .

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Theorem


(i) W is finite.


Sketch of Proof. (i) ⇒ (ii) Thanks to part (b) of the lemmaabove, W acts completely reducibly on V .

Then B must benondegenerate, and the scalars are the only endomorphisms ofV commuting with the action of W .

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Theorem


(i) W is finite.


Sketch of Proof. (i) ⇒ (ii) Thanks to part (b) of the lemmaabove, W acts completely reducibly on V . Then B must benondegenerate, and the scalars are the only endomorphisms ofV commuting with the action of W .

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From part (c) of the lemma above, B is the uniquenondegenerate, W -invariant symmetric bilinear form on V .

But, by part (a) of the lemma, there exists a positive definiteW -invariant form on V , say B ′. so B ′ = cB for some nonzeroc ∈ R. Since B(αs , αs) = 1, we have c > 0. Therefore B is alsopositive definite. �

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From part (c) of the lemma above, B is the uniquenondegenerate, W -invariant symmetric bilinear form on V .But, by part (a) of the lemma, there exists a positive definiteW -invariant form on V , say B ′. so B ′ = cB for some nonzeroc ∈ R.

Since B(αs , αs) = 1, we have c > 0. Therefore B is alsopositive definite. �

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From part (c) of the lemma above, B is the uniquenondegenerate, W -invariant symmetric bilinear form on V .But, by part (a) of the lemma, there exists a positive definiteW -invariant form on V , say B ′. so B ′ = cB for some nonzeroc ∈ R. Since B(αs , αs) = 1, we have c > 0. Therefore B is alsopositive definite. �

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Thanks for your attention

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Radical of the bilinear formDual representationFinite Coxeter groups

Documents

Root systems and Coxeter groups - National Chiao Tung ...weng/courses/2009...and Coxeter groups NCTS Radical of the bilinear form Dual representation Finite Coxeter groups References