根系統和Weyl群的軌跡

» ñ ø ;
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應 用 數 學 系

碩 士 論 文

根系統和

Weyl

群的軌跡

Root System and Orbits of the Weyl Group

研 究 生:

林 采 瑩

指導老師: 蔡 孟 傑 教 授

根系統和Weyl群的軌跡

Root System and Orbits of the Weyl Group

研 究 生

:

林 采 瑩

Student: Tsai-Yin Lin

指導老師: 蔡 孟 傑 教 授 Advisor: Meng-Kiat Chuah

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

In partial Fulfillment of Requirement

For the Degree of Master

In

Applied Mathematics

June 2006

Hsinchu, Taiwan, Republic of China

根系統和Weyl群的軌跡

研 究 生：林采瑩

指導老師：蔡孟傑 教授

國 立 交 通 大 學

應 用 數 學 系

摘要

令g為一個有限維的複半單李代數，而h是g的Cartan子李代數。則g會導 出一個包含許多根的根系統。每個根沿著它自己的超平面又可導出一個 根反射。這些根反射所生成的群叫做Weyl群，這個群在h∗_{有群作用。現} 在給定任意兩個h∗_{的向量，我們的目標是藉由觀察Weyl群的結構，找出} 一個有系統的方法去判斷這兩個向量是否在同一個Weyl群的軌跡裡。對 於An, Bn, Cn, Dn, G2型態的李代數，我們觀察Weyl群作用在歐氏空間的行 為。對於F4型態的李代數，觀察F4的根系統的自同構與D4的根系統的自同 構之間的關係，並藉此用D4的Weyl群去描述F4的Weyl群。

中 華 民 國 九 十 五 年 六 月

Root System and Orbits of the Weyl Group

Student: Tsai-Yin Lin

Advisor: Meng-Kiat Chuah

Department of Applied Mathematics National Chiao Tung University

Abstract

Let g be a finite dimensional complex semisimple Lie algebra with the Cartan subalgebra h. g induces a root system containing roots. Each root gives a reflection with respect to its hyperplane. These reflections generate a group W called Weyl group acting on on h∗_{. Given two vectors, our purpose is to find a systematic method}

to judge if they are in the sane W -orbit by observing the structure of W . For type

An, Bn, Cn, Dn, G2, we study the W -action on Euclidean space. For type F4, observe

the relation between the automorphism of the root system of F4 and it of D4. Then

Acknowledgement

此論文的完成，首先要感謝指導教授蔡孟傑老師，很早就明確地讓我接觸論文題 目，也常常會給我在一些學業上規畫的建議，在課業或研究有疑問時，他也很高興 看到我們去請教他。也要謝謝胡舉卿學姊，他在研究和體育方面都很優秀，在課業 上不但是我和同學的最佳助教，在我參加系排那段時間，也受過他熱心地指導練 球，讓我對排球有全新的認識。 還有我的同學千砡，他待人幽默、辦事明快，總是發自內心地關心、照顧著身 邊的人，很幸運可以和他互相鼓勵直到完成這篇論文。謝謝我的好朋友玫樺，我們 不常聯絡，但他對我的困難常常可以提出有建設性的建議。謝謝羅經凱學長，跟這 個電腦軟體高手在同一間研究室，讓我的疑難雜症往往立即解決，不但省了我不少 麻煩，還常常可以分享好東西。謝謝從大學時期就一直很照顧我的王立中老師，我 跟他學到的只有一點點卻受用至今。謝謝在研一時給我很多幫助的黃大原老師，每 當有問題去找他，他一直都很樂意幫忙。 最後要感謝任何時候都會關心我的家人，沒有他們的支持，我無法有這樣一個單 純、無後顧之憂的生活環境，讓我想補英文想報名考試想找房子都不用操太多心。

目 錄

Abstract (in Chinese) I

Abstract (in English) II

Acknowledgement III

Contents i

1 Introduction 1

2 Mathematical Background 2

3 Group Action 6

4 Equivalence Relation:Classical cases 10

1

Introduction

Let g be a finite dimensional complex simple Lie algebra. Let h be a Cartan subalgebra of g, and ∆ ⊂ h∗ _{a choice of simple roots. It corresponds to a diagram D = Dyn(g),}

whose vertices are the elements of ∆, known as the Dynkin diagram of g . The Dynkin diagram is independent of the choice of h and ∆. Let Φ ⊂ h∗ _{be all the roots.}

Each α ∈ Φ defines a reflection which preserves Φ, and these reflections generate a subgroup W of Aut(Φ), known as the Weyl group. Let X be the set of all assignments of complex numbers to the vertices of D. By Φ, we can identify h∗ _{with X. Namely,}

the element P_∆cαα ∈ h∗ can be represented by the assignment of the numbers {cα}

on the vertices {α} of D. Since W acts on Φ as well as on h∗_{, it also acts on X. In}

this thesis, we study the orbits of the W -action on X.

This thesis is divided into the following sections. In Section 2, we recall the definitions of Cartan subalgebras, root system, simple roots, Dynkin diagram and Weyl group. In Section 3, we introduce some standard actions on Rn_{by S}

n(symmetric

group) and Zn

2 (n-fold product of Z2) as well as their semi-direct product, so that we

can use them to describe the W -action on h∗_{. In Section 4, we present the main result}

2

Mathematical Background

In this section, we start from the definition of Cartan subalgebra. Every complex semisimple Lie algebra g gives a root system by choosing Cartan subalgebra. A root system of a vector space V induces a Weyl group and a simple system so that we can make use of them to define an equivalence relation on V and begin to observe it type by type. Finally, recall the list of all types of complex semisimple Lie algebras.

Definition 2.1 Let g be a Lie algebra, the adjoint representation ad : g → Endg sending X to adX is given by adx(Y ) = [X, Y ], for all X, Y ∈ g.

Definition 2.2 Let g be a complex semisimple Lie algebra. A Lie subalgebra h of g is called a Cartan subalgebra if

(a)h is maximal abelian.

(b) ad_h is simultaneously diagonalizable.(i.e. there exists basis {vi} of g such that

each vi is an eigenvector of adX for all X ∈ h.)

From (b), we can write g as a simultaneous eigenspace decomposition. That is, g = h ⊕M

α∈Φ

gα, where gα = {Y ∈ g| adXY = α(X)Y, ∀X ∈ h}, h = g0.

Here Φ = {α ∈ h∗_{|α 6= 0 and g}

α 6= 0} and α is a function translating X to the

eigenvalue of adX with respect to eigenvector Y . α is linear because of the bilinearity

of the Lie bracket. Therefore, α ∈ Φ ⊆ h∗_{. Φ is called the root system of g and the}

elements of Φ are called roots.

Proposition 2.3 ([4], Proposition2.17, Corollary2.38) Let B( , ) be Killing form on a Lie algebra g , h ⊆ g is the Cartan subalgebra. Let α a root of g with respect to h. Then

(a)There exists Hα ∈ h such that α(H) = B(H, Hα) for all H ∈ h.

(b)If h0 = spanR{Hα|α ∈ Φ}, then h0 is a real form of h such that α|h0 is real on h0

for all α ∈ Φ, hence Φ can be considered as in h∗

Proposition 2.4 ([4], Corollary2.38) Let h0 ⊆ h ⊆ g be defined as above. Then h∗0

is an inner product space over R.

Note that the inner product in h∗

0 is given by (α, β) = B(Hα, Hβ) for all α, β ∈ h∗0,

where Hα, Hβ is the same as Proposition 2.3(a).

Next, we recall the definition of the root system of general real vector space E. By setting E = h∗

0, g determines a root system through its Cartan subalgebra h .

Now let E be an real inner product ( , ). Any nonzero vector α gives a reflection σα

by σα(β) := β −2(β,α)_(α,α)α, for all β ∈ E. For convenient, denote the number 2(β,α)_(α,α)α by

< β, α >. Hence we have σα(β) = β− < β, α > α.

Definition 2.5 A subset Φ of the euclidean space E is called a (reduced)root system in E if

(a) Φ is finite, spans E, and does not contain 0. (b) If α ∈ Φ, the only multiples of α in Φ are ±α. (c) If α ∈ Φ, the reflection σa leaves Φ invariant.

(d) If α, β ∈ Φ, then < β, α >∈ Z

Φ is called irreducible if it cannot be partitioned into the union of two proper subsets such that each root in one set is orthogonal to each root in the other.

In what follows, we see how a root system induces a Weyl group.

Definition 2.6 Let Φ be a root system in E. Define the Weyl group of Φ by

W = W (Φ) = {σα|α ∈ Φ}.

The main question that we want to discuss in this thesis is to study the orbits of Weyl groups. In Section 4, we introduce the root system of every g of each type of complex semisimple Lie algebras. Then

Definition 2.7 Let E be a vector space. A subset ∆ of Φ is called a simple system if

(a) ∆ is a basis of E.

(b) each root β can be written as β = Σα∈∆cαα with integral coefficients cα all

nonnegative or all nonpositive.

The roots in ∆ are called simple roots.

The simple system is a particular basis. We can use the simple system of a root system to draw Dynkin diagram and write numbers on each vertex to represent the elements of h∗_.

Definition 2.8 Let Φ be a root system of rank n, W its Weyl group, ∆ = {α1, ..., αn}

a simple system of Φ. Define the Dynkin diagram of Φ to be a graph having n vertices where the ith vertex denotes the simple root αi. Then set < αi, αj >< αj, αi > edges

between the ith and the jth vertices for all i 6= j. Finally, if there exists any edge between two vertices with different length, add an arrow from the longer to the shorter of the two roots.

Example 2.9 G2

Recall that a complex semisimple Lie algebra induces a root system, hence a Dynkin diagram. We can classify all complex semisimple Lie algebras through their Dynkin diagram. The following theorem shows that they can be exactly classified in several types. Hence we can study the equivalence relation with respect to each type of complex semisimple Lie algebras.

Theorem 2.10 If Φ is an irreducible root system of rank n, its Dynkin diagram is one of the following:

An Bn Cn Dn E6 E7 E8 F4 G2

3

Group Action

To understand the orbit of a Weyl group, we observe the structures of those Weyl groups and described them (on Rn_{) by other groups that we are more familiar in}

experience. Therefore, we need some definition for those group action on Rn _which

describes the Weyl group action on h∗_.

In this article, the elements in Euclidean space are represented by column vector in order to separate from the elements in symmetric groups.

Definition 3.1 Let x =      x1 ... xn      ∈ R

n_{, we define the group action G × R}n _{→ R}n

with respect to the following types of G :

(a) G = Sn (Sn is the symmetric group of degree n) acts on Rn

Define σ.x = σ.      x1 ... xn     =      x_σ−1₍₁₎ ... x_σ−1_(n)     , for σ ∈ G. (b) G = Zn 2 acts on Rn Define b.x =      b1 ... bn     .      x1 ... xn     =      (−1)b1_x 1 ... (−1)bn_x n      , for b =      b1 ... bn     ∈ G. (c) G = Zn−1₂ acts on Rn Define b.x =      b1 ... bn−1     .      x1 ... xn     =         (−1)b1_x 1 ... (−1)bn−1_x n−1 (−1)Σn−11 bi_x n         , for b =      b1 ... bn−1     ∈ G. (d) G = Zn 2 o Sn acts on Rn

Note that the group action in G is defined by (b1, σ1)(b2, σ2) = (b1+ σ1.b2, σ1σ2),

for all (b1, σ1), (b2, σ2) ∈ G. Define (b, σ).x = b.(σ.x), for b ∈ Zn2, σ ∈ Sn, where (σ.x)

is defined in case (a) previously. (e) G = Z2× S3 acts on R3

Define (b, σ).x = (−1)b_{(σ.x), for b ∈ Z}

2, σ ∈ S3, where (σ.x) is defined in case (a)

previously.

Note that the definition in (a) and (b) are special cases of that in (d). They can be obtained by setting identity of the first and the second group of Zn

2o Sn respectively.

Example 3.2 We give some examples for the above definition of group actions. In what follows, the examples (a),(b),(c),(d),(e) correspond respectively to the group actions in Definition 3.1 (a),(b),(c),(d),(e).

(a) Let n = 3, then (1 2 3).      x1 x2 x3     =      x3 x1 x2     . (b) Let n = 3, then      1 0 1     .      5 6 7     =      −5 6 −7     . (c) Let n = 4, then      1 0 1     .         5 6 7 8         =         −5 6 −7 8         . (d) Let n = 3, then           1 0 1     , (1 2)     .      5 6 7     =      1 0 1     .      6 5 7     =      −6 5 −7     .

(e) (1, (2 1 3)) .      5 6 7     =      −6 −7 −5     .

Definition 3.3 Let g be a complex semisimple Lie algebra with Cartan subalgebra h. Let Dyn(g) be the Dynkin diagram of g and fix a simple system {αi}ni=1 of h∗0, we

can denote elements of h∗

0 by writing numbers on the vertices of Dyn(g) corresponding

to the coefficients of the linear combination of ∆.

Example 3.4 Fix a simple system {αi}ni=1 of An, then

a₁ a₂ a_n-1 a_n

denotes Σn i=1aiαi.

Definition 3.5 Let ∆ = {αi}ni=1 be a simple system of a vector space V , W =

({σαi|i = 1, ..., n}) be the Weyl group. Let a = Σaiαi, b = Σbiαi ∈ V we say a is

equivalent to b with respect to W if there exists σ ∈ W such that σa = b, and denote it by      a1 ... an     ∼      b1 ... bn    

 or a ∼ b if we denote a, b by Dynkin diagram.

Example 3.6 Let V = R3 _{with simple system}

∆ = ∆(B3) = {α1 = e1− e2, α2 = e2− e3, α3 = e3}

Consider a vector v = 2α1+ 3α2+ 5α3 . Then

σα3σα2(v) = 2α1+ 4(α2+ 2α3) + 5(−α3) = 6α1+ 4α2+ 3α3 and 3 5 2 ∼ 4 5 2 ∼ 4 3 6 .

The main question that we are curious is if there exists some convenient method to check whether two vectors are equivalent or not. Next, to solve the problem, we are going to observe the equivalence relation on a different basis through some groups isomorphic to the Weyl groups.

4

Equivalence Relation:Classical cases

In Section 4 and 5, we introduce some construction of root systems of complex semisimple Lie algebras where Section 4 is for the classical cases and Section 5 is for the exceptional cases. After observing the action of their Weyl groups on another basis of h∗_{, we will find that the behaviors of those actions are very straight forward.}

To observe the relation between bases and Weyl group action later, it is useful to define some notation to represent a vector with respect to a basis for the discussion.

Definition 4.1 Let V be an n-dimensional vector space over a field F , β = {v1, . . . , vn}, γ =

{u1, . . . , un} be two bases of V .

(a) For x ∈ V , let a1, ...an be the unique scalars such that x = Σni=1aivi.

We define [x]β ∈ Fn by [x]β =      a1 ... an     . (b) Let bij(i, j = 1, . . . , n) be the scalars such that

vj = Σni=1bijui for 1 ≤ j ≤ n.

We define the n × n matrix [1]γ_β by [1]γ_β = (bij).

Now, we are going to observe the root system of each type of complex semisimple Lie algebras. In what follows, we still denote elements in Euclidean space by column vectors written in the forms

     x1 ... xn      or (x1· · · xn) t_. Type An(n ≥ 1)

Consider the hyperplane V =      1 ... 1      ⊥ = {      x1 ... xn+1     |x1+ · · · + xn+1 = 0} ⊆ R n+1_, as well as Φ = {v ∈ V ∩ Zn_|kvk2 _{= 2} = {e} i− ej|1 ≤ i 6= j ≤ n + 1}.

Then Φ is a root system of type An, and

∆ = ∆(An) = {α1 = e1− e2, . . . , αn= en− en+1}

is the simple system of Φ.

In V , since σαi permutes ei, ei+1and leaves all other ej0s fixed, σαi corresponds to

the transposition (i i+1) in the symmetric group Sn+1. These transpositions generate

Sn+1, so we obtain W ∼= Sn+1.

The behavior of W on the standard basis is so simple and direct that we can easily judge if two vectors in V are equivalent with respect to W . Given two vectors

a, b ∈ V , we have [a]std and [b]std in Rn+1 by choosing standard basis to represent

them. To ask if there exists an element of Weyl group translating a to b is equivalent to ask if there exists a permutation of coordinates translating [a]std to [b]std.

There-fore, we have the following result.

Lemma 4.2 a 1 a2 an-1 an ∼ b₁ b₂ b_n-1 b_n if and only if there exists σ ∈ Sn+1 such that σ.M(a1· · · an)t= M(b1· · · bn)t, where

M =            1 −1 1 −1 . .. . .. 1 −1            .

Type Bn(n ≥ 2)

We just follow the idea that we did in type An. Let V = Rn, as well as

Φ = {v ∈ V ∩Zn_|kvk2 _{= 1 or kvk}2 _{= 2} = {±e}

i|1 ≤ i ≤ n}∪{±ei±ej|1 ≤ i ≤ j ≤ n}.

Then Φ is a root system of Bn and

∆ = ∆(Bn) = {α1 = e1− e2, . . . , αn−1= en−1− en, αn = en}

is the simple system of Φ. In V , σαi permutes ei, ei+1 for i = 1, . . . , n − 1, and σαi

changes the sign of en. These generate all permutations and sign changes of

stan-dard coordinates, and can be described by Zn

2oSn. Hence we have the following result.

Lemma 4.3

a₁ a₂ a_n-2 a_n-1 a_n

∼

b

1 b2 bn-2 bn-1 bn

if and only if there exists σ ∈ Zn

2oSnsuch that σ.[1]std∆(Bn)(a1· · · an) t_{= [1]std} ∆(Bn)(b1· · · bn) t_, where [1]std_∆(Bn)=         1 −1 . .. . .. 1 −1 1         n×n .

(i.e. they are different from a permutation and some sign changes in standard coor-dinates representation.)

Type Cn(n ≥ 3)

The case of Cn is almost the same as Bn. Consider V = Rn, then

Φ = {±2ei} ∪ {±ei± ej|1 ≤ i < j ≤ n}

is a root system of type Cn and

is the simple system of Φ. In V , σαi permutes ei, ei+1 for i = 1, . . . , n − 1, and σαi

changes the sign of en. These generate the same group action on standard coordinates

as type Cn. Therefore, the Weyl group action in type Cn is the same as that in type

Bn. Hence we can make use of the same method to judge whether two vectors are

equivalent. Lemma 4.4 a 1 a2 an-2 an-1 an ∼ b 1 b2 bn-2 bn-1 bn if

and only if there exists σ ∈ Zn

2 o Sn s.t. σ.[1]std∆(Cn)(a1· · · an) t _{= [1]}std ∆(Cn)(b1· · · bn) t_, where [1]std ∆(Cn) =         1 −1 . .. . .. 1 −1 2         n×n . Type Dn(n ≥ 4)

Consider V = Rn_{, as well as a root system}

Φ = {±ei± ej|1 ≤ i < j ≤ n}

corresponding to the simple system

∆ = ∆(Dn) = {α1 = e1− e2, . . . , αn−1= en−1− en, αn = en−1+ en}.

In V , σαi permutes ei, ei+1 , for i = 1, . . . , n − 1; σαn permutes en−1, en and changes

their sign simultaneously. These generate all permutations and all sign changes of even number. Such kind of sign changes can be described by Zn−1

2 , since the n-th

component in coordinate is determined by the other n − 1 components. Therefore,

Lemma 4.5 a 1 a n-1 a_n a n-2 a 2 ∼ b 1 b_n-1 b_n b n-2 b 2 if and

only if there exists σ ∈ Zn−1

2 o Sn such that σ.[1]std∆(Dn)(a1· · · an) t _{= [1]std} ∆(Dn)(b1· · · bn) t_, where [1]std ∆(Dn) =         1 −1 . .. . .. 1 1 −1 1         n×n .

5

Equivalence Relation:Exceptional cases

In this section, we keep the same work as Section 4 for exceptional cases of complex semisimple Lie algebras. For type E, we have not find a method good enough to study the orbit of Weyl group yet. So here we only discuss the type F4 and G2 .

Type F4

Let V = R4_{, and the root system}

Φ = {±ei± ej|1 ≤ i < j ≤ 4} ∪ {±ei|i = 1, 2, 3, 4} ∪ {

1

2(±e1± e2± e3± e4)}, as well as the simple system

∆ = {α1 = e2− e3, α2 = e3− e4, α3 = e4, α4 =

1

2(e1− e2− e3− e4)}.

When a vector are represented by standard basis, it become more complex after

σα4 moving it. Hence it is not a good idea to follow the same method to observe the

orbits of W (F4). Instead, we make use of the relation between W (F4) and W (D4).

The relation of the Weyl group of F4 and D4 is associated by the automorphism of

their root system which we are going to discuss.

Definition 5.1 Let Ψ be a root system. Define

Aut(Ψ) = {φ : Ψ → Ψ|φ is linear and < α, β >=< φ(α), φ(β) > for all α, β ∈ Ψ}.

Consider Φ0_{, the root system of D}

4, observe that the 24 long roots in Φ form a root

system Φ0 _{of type D}

4. In what follows, we are going to show that W (Φ) = Aut(Φ0).

Consider W (Φ0_{) ⊂ W (Φ) = Aut(Φ}0_{), other automorphisms of Φ}0 _{arise naturally from}

Aut(Dyn(D4)). Finally, W (Φ) = Aut(Dyn(Φ0)) n W (Φ0) = S3n W (D4). Next, we

Definition 5.2 A lattice is a discrete subgroup of Euclidean space and contains the origin. Define the lattices L1, L2 in Rn:

(a) L1 = {Σni=1aiei ∈ Zn|Σni=1ai is even} is a subgroup of Zn.

(b) L2 = Zn+ Z1₂(Σni=1ei) = {v + k₂Σni=1ei|v ∈ Zn, k ∈ Z}.

Lemma 5.3 Let Φ0 _{be the root system of D}

4 that we have defined previously. Then

(a)Aut(Φ0_{) preserves ( , ) in Φ}0

(b)Aut(Φ0_{) preserves < , > in L}

1

(c)Aut(Φ0_{) preserves < , > in L}

2

Proof. (a) Let φ ∈ Aut(Φ0_{), α, β ∈ Φ}0_{. Then}

2(α, β) (β, β) =< α, β >=< φ(α), φ(β) >= 2(φ(α), φ(β)) (φ(β), φ(β)) . Since (β, β) = (φ(β), φ(β)), (α, β) = (φ(α), φ(β)). (b) Let φ ∈ Aut(Φ0_{), α, β} 1, β2 ∈ Φ, c ∈ R. Then (φ(α), φ(cβ1+ β2)) = (φ(α), cφ(β1) + φ(β2)) = c(φ(α), φ(β1)) + (φ(α), φ(β2)) = c(α, β1) + (α, β2) = (α, cβ1+ β2) .

Since all elements in L1 are linear combination of Φ0, Aut(Φ0) preserves the inner

product (·, ·) of L1. Therefore, it also preserves < , > in L1.

(c) Let φ ∈ Φ0_{, λ ∈ L}

2. It is obvious that 2λ ∈ L1. By (b), we have

< φ(λ), φ(λ) >=< 2φ(λ), 2φ(λ) >=< φ(2λ), φ(2λ) >=< 2λ, 2λ >=< λ, λ >. ¤

Proposition 5.4 Let Φ and Φ0 _{be the root systems of F}

4 and D4 defined as previous

respectively, then Aut(Φ) = Aut(Φ0_).

Proof. Recall that Φ have disjoint three parts:

Φ = {±ei± ej|1 ≤ i < j ≤ 4} ∪ {±ei|i = 1, 2, 3, 4} ∪ {

1

Observe that for all τ ∈ Aut(Φ0_{), τ is stable on these three parts respectively.}

Hence τ is stable on Φ. In addition, based on the last lemma and the fact that Φ ⊆ L2, τ preserves < , > in Φ. It follows that Aut(Φ0) ⊆ Aut(Φ). Conversely,

Aut(Φ) ⊆ Aut(Φ0_{) because the elements in Φ}0 _{are exactly the long roots of Φ. ¤}

The next Corollary is followed by Proposition 5.4 and the fact that

Aut(Φ) = Aut(Dyn(Φ)) n W (Φ).

Corollary 5.5 W (F4) = S3n W (D4)

Proof. Aut(Dyn(F4)) = 1 implies that

Aut(F4) = Aut(Dyn(F4)) n W (F4) = W (F4).

On the other hand,

Aut(D4) = Aut(Dyn(D4)) n W (D4) = S3n W (D4).

Apply Proposition 5.4 , we have W (F4) = S3n W (D4). ¤

Recall that ∆(D4) = {β1 = e1− e2, β2 = e2− e3, β3 = e3 − e4, β4 = e3+ e4} and

Aut(Dyn(D4)) is the set of all bijections of {β1, β3, β4}. Hence Aut(Dyn(D4)) can

naturally be described by S3 consisting of all permutations of {β1, β3, β4}. The group

action Aut(Dyn(D4)) on W (D4) is defined by τ.σβi = σβτ (i) , for all βi ∈ ∆(D4),

for all τ ∈ S3. When [a]∆(F4) ∼F4 [b]∆(F4), it means [a]∆(D4) ∼D4 τ.[b]∆(D4) for some

τ ∈ S3. Now, apply Lemma 4.5, we have the following property.

Lemma 5.6 Let (c1c2 c3 c4)t = [1]_{∆(F 4)}∆(D4)(a1a2 a3 a4)tand (d1 d2d3 d4)t= [1]_{∆(F 4)}∆(D4)(b1b2b3b4)t

Then a₂ a₃ a₄ a₂ b 2 b3 b4 b 2

if and only if τ. c 3 c 4 c₂ c₁ ∼ d 3 d 4 d₂ d₁

, for some τ ∈ S3 denoting

the set of all permutations of {β1, β3, β4} ⊆ ∆(D4)

if and only if σ.([1]std ∆(D4)(τ.(c1 c2 c3 c4) t_{)) = [1]}std ∆(D4)(d1 d2 d3 d4) t_{, for some σ ∈} S4n Z32, some τ ∈ S3, where [1]∆(D4) ∆(F4) =         0 0 0 1 2 1 0 0 0 0 1 −1 2 0 0 0 1 2 − 1 2         . Example 5.7 Let (a b c d)t_{= [1]}∆(D4) ∆(F 4)(x y z t)t, (a b d c)t= [1] ∆(D4) ∆(F 4)(x0 y0 z0 t0)t and ∆(D4) = {β1, β2, β3, β4}. Then ((1 3), σβ3).      c d b a     = σβ3       a d b c      = -a d b+a c ((1 4 3), σβ1).      -a d b+a c     = σβ1       d c b+a -a      = d c b a = ((3 4), 1).      c d b a     

This corresponds to the operation in S3n W (D4), i.e. ((1 4 3), σβ1)((1 3), σβ3) =

c d b a ∼ d c b a and x y z t ∼ x’ y’ z’ t’ . Type G2 Consider V =      1 1 1      ⊥ = {      x1 x2 x3     | x1+ x2+ x3 = 0} ⊆ R 3_. Define Φ = {v ∈ V ∩ Zn_|kvk2 _{= 2 or kvk}2 _{= 6}}

= {±(ei− ej)|1 ≤ i < j ≤ 3} ∪ {±(2ei− ej− ek)|{i, j, k} = {1, 2, 3}}

= {±(e1− e2), ±(e2− e3), ±(e1− e3), ±(2e1 − e2− e3), ±(2e2− e1 − e3), ±(2e3− e1− e2)}

Then Φ is a root system of type G2, and

∆ = {α1 = e1− e2, α2 = −2e1+ e2+ e3}

is the simple system of Φ.

In V , σα1 permutes e1, e2; σα2 permutes e2, e3 and changes sign of any vector

in V . Recall that the definition of group action of Z3

2 o S3 on R3, we have σα1 =

((0 0 0)t_{, (1 2)) and σ}

α2 = ((1 1 1)t, (2 3)) in V . Since W is generated by σα1 and σα2, W has an embedding in Z32o S3. That is,

W =< σα1, σα2 >∼=<

¡

(0 0 0)t_{, (1 2)}¢_,¡_{(1 1 1)}t_{, (2 3)}¢_>

(In this thesis, < · > in a Weyl group always means the group generation).

the additive group < (0 0 0)t_{, (1 1 1)}t _{>∼= Z}

2 which acts on V and denotes sign

change(simultaneously sign change on all standard coordinates of V). Therefore,

W ∼=< (0, (1 2)), (1, (2 3)) >= Z2× S3.

In fact,

W ∼= D6 = {σ0, . . . , σ5, τ, στ, . . . , σ5τ }

with |σ| = 6, τ2 _{= 1, and στ σ = τ , where σ = (1, (1 2 3)), τ = (0, (1 2)). We make use}

of the former structure (Z2× S3) in that it is more helpful to check the equivalence

relation for vectors in V .

Considering all the factors above, for two vectors a, b in V, [a]∆(G2) ∼ [b]∆(G2)

means that the difference between [a]std and [b]std in R3 are their arrange or sign.

Hence we have the following result.

Lemma 5.8

a b

∼

c d

if and only if ∃σ ∈ Z2 × S3 such that

σ.M¡a_b¢ = M¡_dc¢, where M =      1 −2 −1 1 0 1     .

Corollary 5.9 Let x ∈ V , suppose that x = aα1+ bα2 = x1e1+ x2e2+ x3e3, for some

scalars a, b, x1, x2, x3 in R. Then the equivalence class of x is

參 考 文 獻

[1] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York 1972.

[2] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1990.

[3] N. Bourbaki, Groupes et alg`ebres de Lie, Masson, Paris, 1981.

[4] Anthony W. Knapp, Lie Groups Beyond an Itroduction, 2nd edn., Boston :Birkhauser, 2002.