從代數觀點研究亮點西格瑪遊戲

博士論文

從代數觀點研究亮點西格瑪遊戲

Lit-only sigma-game

from the view of algebra

研究生: 黃皜文

指導教授: 翁志文

從代數觀點研究亮點西格瑪遊戲

Lit-only sigma-game

from the view of algebra

研究生: 黃皜文

Student: Hau-wen Huang

指導教授: 翁志文

Advisor: Chih-wen Weng

國立交通大學

應用數學系

博士論文

A Thesis

Submitted to Department of Applied Mathematics

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

in

Applied Mathematics

July 2011

Hsinchu, Taiwan

中華民國一百年七月

研究生: 黃皜文

指導教授: 翁志文

國立交通大學

應用數學系

摘要

Lit-only sigma-game

from the view of algebra

Student: Hau-wen Huang

Advisor: Chih-wen Weng

Department of Applied Mathematics

National Chiao Tung University

Abstract

The lit-only σ-game is a one-player game played on a finite simple graph. It is known that this game can be view as a group action. In this thesis we show how this game is related to Coxeter groups. Moreover we use algebraic techniques to generalize some known results on the game.

完成此論文，我想感謝很多人，最感謝是我的阿媽。從小把我帶大，

雖然阿媽不識半字，但阿媽懂得加減法、看時鐘。我相信我的數學細胞

都是遺傳自於阿媽。謝謝我兩位姐姐、姑姑、雙親等人給予我生活上的

幫助。感謝我的女友及其家人，謝謝他們的支持和生活上的幫忙。

在我求學過程中，國中時期的數學老師鄭瑞欽是我數學上的啟蒙教

師，一堂又一堂有趣且縝密的數學課，讓我開始對數學有了初步了解。

大學時期於清華大學數學系所修讀的課程，使我對數學有進一層的認

識，在此感謝所有教導過我的任課教授。謝謝我碩博班的指導教授翁志

文，謝謝他協助我撰寫此論文、給予研究費讓我經濟無虞、以及協助我

前往麥迪遜威斯康辛大學數學系接觸不同的研究主題。

最後，謝謝同研究室的學長李信儀、楊川和及學弟陳德軒平常的照

顧。謝謝以前碩班同學張澍仁、李張圳、陳柏澍、卜文強、張雁婷等人

平常的照顧。

1 Introduction 1 2 Lit-only sigma-game and simply-laced Coxeter groups 3

2.1 The flipping group of a graph . . . 3

2.2 A representation of the Coxeter group of type Γ . . . 5

2.3 The center of the flipping group W of type Γ . . . . 6

2.4 Lit-only σ-game on the Dynkin diagram of type An . . . 7

2.5 Lit-only σ-game on the Dynkin diagram of type Dn . . . 9

2.6 Lit-only σ-game on Γ and its induced subgraph . . . 11

2.7 Lit-only σ-game on the Dynkin diagram of type En . . . 12

2.8 Summary . . . 16

3 Lit-only sigma-game on a graph with a long induced path 19 3.1 The sets Π, Π0 and Π1 . . . 20

3.2 The simple basis ∆ of FS 2 . . . 22

3.3 The case |Π1| is odd . . . 22

3.4 The case _|Π1| is even . . . 25

3.5 Summary . . . 28

3.6 Remarks . . . 29

4 One-lit trees for lit-only sigma-game 31 4.1 The degenerate and nondegenerate graphs . . . 31

4.2 Some combinatorial properties of nondegenerate trees . . . 32

4.3 The Reeder’s game . . . 32

4.4 Reeder’s game on a nondegenerate tree . . . 34

4.5 Lit-only σ-game on a nondegenerate tree . . . 36

4.6 A homomorphism between simply-laced Coxeter groups . . . 38

4.7 More one-lit trees for lit-only σ-game . . . 40

4.8 Combinatorial statements of Theorems 4.5.7 and 4.7.7 . . . 43

5 The edge-version of lit-only sigma-game 45 5.1 The edge space and the bond space . . . 45

5.2 The edge-flipping group of Γ . . . 46

5.3 The structure of WR in the case Γ is a tree . . . 47

5.4 The WR-orbits of R . . . 48

5.5 The minimum light number for e-lit-only σ-game on Γ . . . 50

Chapter 1

Introduction

My object of this thesis is to use algebraic techniques to study a combinatorial game called the lit-only σ-game. The game is a one-player game played on a finite graph. Let Γ denote a finite graph. A configuration of the lit-only σ-game on Γ is an assignment of one of two states, on or off, to each vertex of Γ. Given a configuration, a move of the lit-only

σ-game on Γ allows the player to choose one on vertex s of Γ and change the states of all

neighbors of s. Given a starting configuration, the goal is usually to minimize the number of on vertices of Γ or to reach an assigned configuration by a finite sequence of moves. In the thesis, we are only concerned with the lit-only σ-game on a finite simple graph and always assume that Γ is a finite simple graph.

The game implicitly appeared in the classification of simple Lie algebras over real number field. See [2, 8] for details. In 2005 International and Third Cross-strait Confer-ence on Graph Theory and Combinatorics, Gerard J. Chang’s talk “Graph Painting and Lie Algebra” promoted the birth of this game. Later Yaokun Wu and Xinmao Wang [26] realized this game is a variation of σ-game and named it lit-only σ-game. They also found that the game appeared as early as 2001 in the paper [12].

As far as we know, the first result on this topic is from [2], which claimed that if Γ is a simply-laced Dynkin diagram then given any configuration one can reduce the number of on vertices to at most one. Some results of [8] can be viewed as a description of the orbits of this game on simply-laced Dynkin diagrams. Gerard J. Chang, on his talk, gave a conjecture: if Γ is a tree with ℓ leaves then for any configuration one can reduce the number of on vertices to at most _⌈ℓ

2⌉. Later Yaokun Wu and Xinmao Wang [26] proved

this conjecture. Also they [26] found that a subgroup of the general linear group over the two-element field of which the natural action can be viewed as the lit-only σ-game. Later in the paper [29], Yaokun Wu named this group the lit-only group and proved that it is isomorphic to the symmetric group on n letters when the underlying graph is the line graph of a tree of order n_{≥ 3. In 2007 the author independently found this group, and in} 2008 the author named it the flipping group. In this dissertation we will adopt the latter name. For the study of the difference between the lit-only σ-game and σ-game, please refer to [14, 15, 27].

The organization of this dissertation is as follows. In Chapter 2 we show how the flipping groups are related to the simply-laced Coxeter groups, and from the view of the flipping groups we give an alternative description of the orbits of the game on simply-laced Dynkin diagrams. In Chapter 3 we consider the game on an n-vertex graph with an

induced path of n_{− 1 vertices, which generalizes the study of the latter part of Chapter} 2. Motivated by the first result [2], Chapter 4 is devoted to finding more trees for which given any configuration one can reach a configuration with at most one on vertex by a finite sequence of moves. The topic of Chapter 5 is to study the edge-version of lit-only

σ-game on Γ. We may view this variation as the lit-only σ-game on the line graph L(Γ)

of Γ. We find that the structure of the flipping group of L(Γ), which only depends on the order and size of Γ.

Chapter 2

Lit-only sigma-game and

simply-laced Coxeter groups

The lit-only σ-game is a one-player game played on a finite simple graph. Let Γ denote a finite simple graph. A configuration of the lit-only σ-game on Γ is an assignment of one of two states, on or off, to all vertices of Γ. Given a configuration, a move of the lit-only

σ-game on Γ consisting of choosing one on vertex s of Γ and changing the states of all

neighbors of s. Given a starting configuration, the goal is usually to minimize the number of on vertices of Γ or to reach an assigned configuration by a finite sequence of moves. In this chapter, we show how the lit-only σ-game is related to simply-laced Coxeter groups and study the game on simply-laced Dynkin diagrams.

2.1

The flipping group of a graph

An ordered pair Γ = (S, R) is called a finite simple graph whenever S is a finite set and R is a set of some two-element subsets of S. The elements of S are called vertices of Γ and the elements of R are called edges of Γ. For any s, t_{∈ S we say s and t are neighbors} whenever {s, t} ∈ R. For convenience we usually write st ∈ R or ts ∈ R for {s, t} ∈ R. We say that a finite simple graph Γ = (S, R) is connected whenever for any two distinct vertices s, t of Γ there exists a subset_{s0s1, s1s2, . . . , sk−1sk} of R with s0 = s and sk= t.

Throughout this dissertation let Γ = (S, R) denote a finite simple graph. Moreover we assume that S is nonempty and that Γ is connected. Let_F2 denote the two-element field

{0, 1}. Let MatS(F2) denote the set consisting of square matrices over F2 with rows and

columns indexed by S. Let GLS(F2) denote the group consisting of all invertible matrices

in MatS(F2). The group operation of GLS(F2) is ordinary matrix multiplication. We use

I to denote the identity in GLS(F2). Let FS2 denote the vector space consisting of column

vectors over _F2 indexed by S. For s∈ S let es denote the characteristic vector of s inFS2;

i.e. es = (0, 0, . . . , 0, 1, 0, . . . , 0)t, where 1 is in the position corresponding to s. Here at

means the transpose of a.

We interpret each configuration a of the lit-only σ-game on Γ as the vector ∑

s

of _FS

2, where the sum is over all vertices s of Γ that are assigned the on state by a; if all

vertices of Γ are assigned the off state by a, then (2.1) is interpreted as zero vector. We may view a move of the lit-only σ-game as choosing any vertex s of Γ and changing the states of all neighbors of s if the state of s is on.

Definition 2.1.1. For s_{∈ S define a matrix κ}s∈ MatS(F2) by

(κs)uv=

{

1 if u = v, or v = s and uv ∈ R,

0 else

for all u, v_{∈ S.}

The following is a reformulating of Definition 2.1.1.

Lemma 2.1.2. For s, v _{∈ S we have}

κsev = { ev+ ∑ uv∈R eu if v = s, ev if v̸= s. Let a _{∈ F}S

2. By Lemma 2.1.2, if the state of s is on then κsa is obtained from a by

changing the states of all neighbors of s; if the state of s is off then κsa = a. Therefore

we may view κs as the move of the lit-only σ-game on Γ for which we choose the vertex

s and change the states of all neighbors of s if the state of s is on.

Lemma 2.1.3. For s_{∈ S we have κ}2

s = I. In particular κs ∈ GLS(F2).

Proof. Use Lemma 2.1.2.

Definition 2.1.4. Let W denote the subgroup of GLS(F2) generated by κs for all s∈ S.

We call W the flipping group of Γ.

As far as we know the flipping group of Γ was first mentioned in [26, Introduction]. Observe that for any a, b _{∈ F}S

2, b is obtained from a by a finite sequence of moves

of the lit-only σ-game on Γ if and only if b = Ga for some G ∈ W. We now define the

W-orbits of FS

2, which are exactly the orbits of the lit-only σ-game on Γ.

Definition 2.1.5. Let a∈ FS

2. By the W-orbit of a we mean the set Wa ={Ga | G ∈ W}.

By a W-orbit of _FS

2 we mean a W-orbit of a for some a∈ FS2.

We finish this section with a property about the flipping group W of Γ. To see this we establish a lemma.

Lemma 2.1.6. For s_{∈ S define E}s ∈ MatS(F2) by

Esev =

{

0_∑ if v̸= s,

uv∈R

eu if v = s. (2.2)

for all v ∈ S. Then the following (i)–(iii) hold.

2.2. A representation of the Coxeter group of type Γ (ii) EsEt= 0 if st /∈ R.

(iii) If sisi−1 ∈ R for i = 1, 2, . . . , k then

EskEsk−1· · · Es0 =

{

Es0 if sk = s0, EskEs0 if sks0 ∈ R.

Proof. (i) is immediate from Lemma 2.1.2. Using (2.2) we find EsEtev = 0 for any

v, s, t ∈ S with st ̸∈ R. Hence we have (ii). (iii) follows from the same reason as in (ii)

by applying the product of matrices in either side of the equation to ev and obtaining the

desired equality in each case.

Proposition 2.1.7. For s, t∈ S we have (κsκt)2 = I if st̸∈ R and (κsκt)3 = I if st∈ R.

Proof. By Lemma 2.1.6(i)

κsκt= (I + Es)(I + Et)

= I + Es+ Et+ EsEt.

In the case s̸= t and st ̸∈ R,

(κsκt)2 = (I + Es+ Et)(I + Es+ Et)

= I + 2Es+ 2Et

= I by Lemma 2.1.6(ii). In the case st∈ R,

(κsκt)2 = (I + Es+ Et+ EsEt)(I + Es+ Et+ EsEt) = I + 3Es+ 3Et+ 4EsEt+ EtEs = I + Es+ Et+ EtEs and (κsκt)3 = (κsκt)2(κsκt) = (I + Es+ Et+ EtEs)(I + Es+ Et+ EsEt) = I + 2Es+ 4Et+ 2EsEt+ 2EtEs = I by Lemma 2.1.6(iii).

2.2 A representation of the Coxeter group of type Γ

A Coxeter group is a group generated by a set T subject to relations of the form (st)m(s,t)= 1 for all s, t∈ T ,

where m(s, s) = 1 and m(s, t) = m(t, s)_{∈ {2, 3, . . . , ∞} for s ̸= t in T. If m(s, t) ∈ {2, 3}} for all s_{̸= t in T, the Coxeter group is said to be simply-laced. Proposition 2.1.7 motivates} us to consider a certain (simply-laced) Coxeter group as follows.

Definition 2.2.1. Let W denote the group generated by all elements of S subject to the

following relations

s2 = 1, (st)2 = 1 if st̸∈ R, (st)3 = 1 if st∈ R for all s, t∈ S. We call W the (simply-laced) Coxeter group of type Γ.

We now establish a connection between the Coxeter group of type Γ and the lit-only

σ-game on Γ.

Theorem 2.2.2. There exists a unique representation κ : W _{→ GL}S(F2) such that

κ(s) = κs for all s∈ S. In particular κ(W ) = W.

Proof. Immediate from Proposition 2.1.7 and Definition 2.2.1.

For the rest of this dissertation let κ denote as in Theorem 2.2.2.

For the rest of this chapter we shall give a new description of W-orbits of _FS

2 when Γ

is a simply-laced Dynkin diagram, which is different than the description from [8].

An(n≥ 1) c c c q q q c c c sn sn−1 sn−2 s3 s2 s1 Dn(n≥ 4) c c c c q q q c c c "" bb sn−1 sn sn−2 sn−3 s3 s2 s1 E6 c c c c c c s5 s4 s3 s2 s1 s6 E7 c c c c c c c s6 s5 s4 s3 s2 s7 s1 E8 c c c c c c c c s7 s6 s5 s4 s3 s8 s2 s1 Figure 1.1: simply-laced Dynkin diagrams.

2.3

The center of the flipping group W of type Γ

Proposition 2.3.1. Let Z(W) denote the center of W. Then Z(W) ={I}.

Proof. Let G denote any element in Z(W) and let u, v denote two distinct elements

in S. We show that the (v, u)-entry Gvu of G is zero to conclude G = I. Proceed by

contradiction. Suppose Gvu = 1. On the one hand κvG eu ̸= Geu since Geu has 1 in the

vth position. On the other hand, κvG eu = Gκveu = Geu since κveu = eu. Hence we have

2.4. Lit-only σ-game on the Dynkin diagram of type An

Corollary 2.3.2. Let Z(W ) denote the center of W. Then Z(W ) is contained in the

kernel of κ.

Proof. Immediate from Proposition 2.3.1.

Since the generator s _{∈ S have order 2 in W, each w ̸= 1 in W can be written in} the form w = s1s2· · · sr for some si in S. If r is as small as possible, call it the length

of w. If W has finite order, it is well-known that there exists a unique longest element in W (for example see [21, p. 115]). We shall denote this by w_◦. It is well-known that Z(W ) ={1, w_◦} or {1} (for example see [21, p. 132]).

2.4

Lit-only σ-game on the Dynkin diagram of type

A

In this section we assume that Γ is the (simply-laced) Dynkin diagram of type An

(n ≥ 1). The goal of this section is to show Kerκ = Z(W ) and to determine when κ is irreducible. We also find a description of the W-orbits of_FS

2. We start with the smallest

case n = 1.

Proposition 2.4.1. Assume n = 1. Then the following (i)–(iii) hold.

(i) The W-orbits of _FS

2 are {0}, {1}.

(ii) Ker κ and Z(W ) are equal to _{{1, w◦}.} (iii) The representation κ is irreducible.

Proof. In this case W ={1, s1} and W = {I}. By these (i)–(iii) follow.

For the rest of this section we assume n_{≥ 2. Let}

1 = es1, i + 1 = κsiκsi−1· · · κs11 (1≤ i ≤ n). (2.3)

Note that

i = esi−1 + esi (2≤ i ≤ n), (2.4)

n + 1 = esn = 1 + 2 +· · · + n. (2.5)

Let ∆ = ∆(An) := {1, 2, . . . , n}. Using (2.4) we find that ∆ is a basis of FS2. We refer

∆ to the simple basis of FS₂. For a ∈ FS₂, let ∆(a) denote the subset of ∆ consisting of all the elements appeared in the expression of a as a linear combination of elements in ∆. For a ∈ FS

2 let ||a||s := |∆(a)| and we call ||a||s the simple weight of a. For example

∆(n + 1) = ∆ and||n + 1||s = n.

Lemma 2.4.2. For 1_{≤ i ≤ n, κ}sii = i + 1, κsii + 1 = i and κsi fixes other vectors in {1,

2, . . . , n + 1} \ {i, i + 1}.

Proof. Use Lemma 2.1.2, (2.3), (2.4) to check.

For the rest of this section let Sn+1 denote the symmetric group on {1, 2, . . . , n + 1}.

Definition 2.4.3. Let α : W_{→ S}n+1 denote the homomorphism defined by

α(G)j := Gj (1≤ j ≤ n + 1) for G∈ W.

Note that α(κsi) is the transposition (i, i + 1) in Sn+1 for each 1≤ i ≤ n.

Lemma 2.4.4. α is an isomorphism from W to Sn+1.

Proof. α is surjective since the transpositions α(κs1), α(κs2),. . ., α(κsn) generate Sn+1.

Since ∆_{∪ {n + 1} spans F}S

2, α is injective. The result follows.

Proposition 2.4.5. The W-orbits of FS

2 are

Oi ={a ∈ FS2 | ||a||s = i or n + 1− i} (0≤ i ≤ ⌊n+1₂ ⌋),

where _{⌊t⌋ is the largest integer less than or equal to t.} Proof. Suppose a∈ FS

2 with ||a||s = i. Observe that from Lemma 2.4.4 and (2.5),

∆(Ga) = {

α(G)∆(a) if n + 1̸∈ α(G)∆(a), ∆\ α(G)∆(a) if n + 1_{∈ α(G)∆(a)}

for G ∈ W. The proposition follows from this observation because the subgroup of

α(W) = Sn+1 generated by the transpositions α(κs1), α(κs2),. . ., α(κsn−1) acts

transi-tively on the fixed size subsets of ∆, and κsnn = 1 + 2 +· · · + n by Lemma 2.4.2 and

(2.5).

Proposition 2.4.6. The representation κ is irreducible if and only if n is even.

Proof. Let V denote a nontrivial proper subspace ofFS

2 such that κ(W )V ⊆ V . Referring

to Proposition 2.4.5, note that

V =∪

i∈J

Oi (2.6)

for some proper subset J _{⊆ {0, 1, . . . , ⌊}n+1

2 ⌋} with J ̸= {0}. Note that the set in the

right-hand side of (2.6) to be closed under addition is when it is the set of even weight vectors, and this occurs if and only if n is odd.

Proposition 2.4.7. The representation κ is faithful.

Proof. Immediate from Lemma 2.4.4 and the fact that W is isomorphic to Sn+1 (for

example see [21, p. 41]).

Proposition 2.4.8. Ker κ = Z(W ) is the trivial group.

Proof. By Proposition 2.4.7 Ker κ =_{{1}. By this and Corollary 2.3.2 Kerκ = Z(W ). The}

2.5. Lit-only σ-game on the Dynkin diagram of type Dn

2.5 Lit-only σ-game on the Dynkin diagram of type

D

In this section we assume that Γ is the (simply-laced) Dynkin diagram of type Dn

(n_{≥ 4). We shall do the same things as Section 2.4 for this case.} Let 1 = es1, i + 1 = κsiκsi−1· · · κs11 (1≤ i ≤ n − 1), n + 1 = esn. (2.7) Note that i = esi−1 + esi (2≤ i ≤ n − 2), (2.8) n− 1 = esn−2 + esn−1 + esn, (2.9) n = esn−1 + esn = 1 + 2 +· · · + n − 1. (2.10)

Set ∆ = ∆(Dn) := {1, 2, . . . , n − 1, n + 1} to be the simple basis of FS2 (in the case of

type Dn). For a ∈ FS2 set ∆(a) and||a||s as Section 2.4. For example ∆(n) = ∆\ {n + 1}

by (2.10), and _||n||s= n− 1.

Lemma 2.5.1. The following (i), (ii) hold.

(i) For 1≤ i ≤ n − 1, κsii = i + 1, κsii + 1 = i, and

κsij = j for j ∈ {1, 2, . . . , n + 1} \ {i, i + 1}.

(ii) κsnn− 1 = n, κsnn = n− 1, κsnn + 1 = n− 1 + n + n + 1, and

κsnj = j for j ∈ {1, 2, . . . , n − 2}.

In particular n + 1_{∈ ∆(Gn + 1) and G({1, 2, . . . , n}) ⊆ {1, 2, . . . , n} for all G ∈ W.} Proof. Use Lemma 2.1.2, (2.7)–(2.9) to check.

For the rest of this section let Sn denote the group of permutations on {1, 2, . . . , n}.

By Lemma 2.5.1 we may make the following definition.

Definition 2.5.2. Let β : W→ Sn denote the homomorphism defined by

β(G)(j) = Gj (1≤ j ≤ n) for G∈ W.

Lemma 2.5.3. β : W→ Sn is an epimorphism.

Proof. It follows that the n_{−1 transpositions β(κ}s1), β(κs2), . . . , β(κsn−1) generate Sn.

Let O denote a subset of _FS

2. We say that O is closed under W whenever WO⊆ O.

Proposition 2.5.4. Let Z denote the subspace of_FS

2 spanned by the set {1, 2, . . . , n − 1}.

Proof. Note that a _{∈ Z if and only if n + 1 ̸∈ ∆(a) for a ∈ F}S

2. By Lemma 2.5.1 and

(2.10), Z is closed under W.

Corollary 2.5.5. The representation κ is not irreducible.

Proof. Immediate from Proposition 2.5.4

For the rest of this section let Z denote as in Proposition 2.5.4. By Proposition 2.5.4,

Z is a disjoint union of some W-orbits of FS

2. It follows that FS2 \ Z is also a disjoint

union of some W-orbits of FS

2. To find the W-orbits of FS2, we may divide this into the

two cases: (i) the W-orbits of _FS

2 in Z; (ii) the W-orbits of FS2 inFS2 \ Z.

Proposition 2.5.6. The W-orbits of FS

2 are

Oi ={a ∈ Z | ||a||s= i or n− i} (0≤ i ≤ ⌊n₂⌋),

Ωo ={a ∈ FS2 \ Z | ||a||s≡ 1 or n − 1 (mod 2)},

Ωe ={a ∈ FS2 \ Z | ||a||s≡ 0 or n (mod 2)}.

In particular Ωo = Ωe =FS2 \ Z when n is odd.

Proof. The proof is similar to the proof of Proposition 2.4.5. The reason that Oi is a

W-orbit ofFS

2 follows from two facts: (i) β(κs1), β(κs2), . . . , β(κsn−2) generate the subgroup

Sn−1 of Sn consisting of permutations on ∆\ {n + 1} and Sn−1 acts transitively on fixed

size subsets of ∆_{\ {n + 1}; (ii)}

κsn−1n− 1 = κsnn− 1 = n = 1 + 2 + · · · + n − 1

by Lemma 2.5.1(i), (ii) and (2.10). The reason that Ωo and Ωe are orbits follows from an

additional fact that _||κsnn + 1||s=||1 + 2 + · · · + n − 2 + n + 1||s = n− 1.

From now on we view Z as an additive group. Let Aut(Z) denote the group consisting of all automorphisms of Z. We now study the structure of W.

Definition 2.5.7. Let γ : W→ Aut(Z) denote the homomorphism defined by

γ(G)(u) = Gu

for u_{∈ Z and G ∈ W.}

Lemma 2.5.8. There exists a unique homomorphism θ : Sn→ Aut(Z) such that γ = θ◦β.

Proof. Since β is surjective, it suffices to show that the kernel of β is contained in the

kernel of γ. Suppose G_{∈ Kerβ. Then Gi = i for 1 ≤ i ≤ n. It follows that G fixes each} element of Z. Therefore G∈ Kerγ. The result follows.

In view of Lemma 2.5.8 we can define the (external) semidirect product of Z and Sn

with respect to θ (for example see [23, p.155]). We denote this group by Z oθSn. This

group is the set Z× Sn with the group operation defined by

(u, σ)(v, κ) = (u + θ(σ)(v), σκ),

where u, v _{∈ Z and σ, κ ∈ S}n. Note that n + 1 + Gn + 1∈ Z for any G ∈ W by Lemma

2.6. Lit-only σ-game on Γ and its induced subgraph

Definition 2.5.9. Let δ : W_{→ Z o}θSn denote the map defined by

δ(G) = (n + 1 + Gn + 1, β(G))

for G_{∈ W.}

Lemma 2.5.10. The map δ : W_{→ Z o}θSn is a group monomorphism.

Proof. For G, H ∈ W, δ(G)δ(H) = (n + 1 + Gn + 1, β(G))(n + 1 + Hn + 1, β(H)) = (n + 1 + Gn + 1 + θ(β(G))(n + 1 + Hn + 1), β(G)β(H)) = (n + 1 + Gn + 1 + G(n + 1 + Hn + 1), β(G)β(H)) = (n + 1 + GHn + 1, β(GH)) = δ(GH).

This shows that δ is a homomorphism. Let G _{∈ Kerδ. Since Gn + 1 = n + 1 and G ∈} Ker β, G fixes all vectors in ∆ and so G = I. This shows that δ is injective. The result follows.

Note that Z = n + 1 + Ωo if n is odd, and Z = (n + 1 + Ωo)∪ (n + 1 + Ωe) if n is even.

Lemma 2.5.11. δ(W) = (n + 1 + Ωo)oθSn. Moreover δ(W) = ZoθSn if n is odd, and

δ(W) has index 2 in ZoθSn if n is even.

Proof. Note that δ(κs1), δ(κs2), . . . , δ(κsn−1) generate {0} oθSn. By this and since Ωo is

an orbit containing n + 1, it follows that δ(W) = (n + 1 + Ωo)oθSn. The second part

follows from Proposition 2.5.6.

Proposition 2.5.12. The representation κ is faithful when n is odd; Ker κ has order 2

when n is even. Moreover Ker κ = Z(W ).

Proof. Note that W is isomorphic to the semidirect product Z o Sn of Z and Sn (for

example see [21, p.42]). By Lemma 2.5.11, κ is faithful when n is odd, and Ker κ has order 2 when n is even. From Corollary 2.3.2, Z(W ) _{⊆ Kerκ, and from the fact that a} normal subgroup of order 2 is contained in the center, we have Ker κ⊆ Z(W ).

2.6 Lit-only σ-game on Γ and its induced subgraph

To help us study Ker κ in the case E8, we now discuss some relations between the

lit-only σ-game on Γ and an induced subgraph of Γ.

Let J _{⊆ S. Let W}J denote the subgroup of W generated by the κs for all s∈ J. Let

WJ denote the subgroup of W generated by s∈ J. It is well known that WJ is isomorphic

to the Coxeter group of type Γ[J] (For example see [21, Section 5.5]). Therefore we will use the same symbol WJ to express these two isomorphic groups. For G ∈ MatS(F2) let

G[J ] denote the submatrix of G with rows and columns indexed by J.

Lemma 2.6.1. Let the notation be as above. Let Γ[J] denote the subgraph of Γ induced by

J. Let WJ[J ] denote the set of those G[J ]∈ GLJ(F2) where G∈ WJ. Then the following

(i) WJ[J ] is the flipping group of Γ[J ].

(ii) The map ψ : WJ → WJ[J ] defined by

ψ(G) = G[J ] for G∈ WJ

is a surjective homomorphism.

Proof. By Definition 2.1.1, (κs)uv = 0 for s, u ∈ J and v ∈ S \ J. By this, each matrix

G∈ WJ has the form

G =

(

A 0 B C

)

if indices in J are placed in the beginning of rows and columns, where A is a _{|J| × |J|} matrix, B is an (n_{− |J|) × |J| matrix, C is an (n − |J|) × (n − |J|) matrix, and 0 is a}

|J| × (n − |J|) zero matrix. Then (i), (ii) follows from the following matrix product rule

in block form: ₍ A 0 B C ) ( A′ 0 B′ C′ ) = ( AA′ 0 BA′ + CB′ CC′ ) .

By Theorem 2.2.2 there exists a unique representation κ′ : WJ → GLJ(F2) such that

κ′(s) = κs[J ] for all s∈ J.

Lemma 2.6.2. Let the notation be as above. Then the following (i), (ii) hold.

(i) κ′ = ψ◦ κ  WJ.

(ii) Ker κ WJ ⊆ Kerκ′.

Proof. Since (ψ◦ κ)(s) = κs[J ] = κ′(s) for all s∈ J, it follows that κ′ = ψ◦ κ  WJ. This

shows (i). (ii) immediate from Lemma 2.6.1(i) and (i).

2.7 Lit-only σ-game on the Dynkin diagram of type

E

In this section we assume that Γ is the graph in Figure 1.2. We shall give a description of W-orbits of FS

2. Restricting to the case n = 6, 7, 8, we shall show that Ker κ = Z(W ).

En(n≥ 6) c c c c c c q q q c c c sn−1 sn−2 sn−3 sn−4 sn−5 sn s3 s2 s1 Figure 1.2: a finite simple graph E_n

Let 1 = es1, i + 1 = κsiκsi−1· · · κs11 for 1≤ i ≤ n − 1 and n + 1 = esn. Note that

i = esi + esi−1 (2≤ i ≤ n − 3),

n− 2 = esn−3 + esn−2 + esn, (2.11)

n− 1 = esn−2 + esn−1 + esn,

2.7. Lit-only σ-game on the Dynkin diagram of type En

Set ∆ = ∆(En) := {1, 2, . . . , n} to be the simple basis of FS2 in this case. Observe that

n + 1 = 1 + 2 +· · · + n. (2.12)

Set ∆(a) and ||a||s = |∆(a)| as before for a ∈ FS2. For example ∆(n + 1) = ∆ and

||n + 1||s= n.

Lemma 2.7.1. The following (i), (ii) hold.

(i) For each 1≤ i ≤ n − 1, κsii = i + 1, κsii + 1 = i, and

κsij = j for j ∈ {1, 2, . . . , n + 1} \ {i, i + 1}.

(ii) κsnn + 1 = n− 2 + n − 1 + n, κsnn = n− 2 + n − 1 + n + 1, κsnn− 1 = n − 2 +

n + n + 1, κsnn− 2 = n − 1 + n + n + 1 and

κsnj = j for 1≤ j ≤ n − 3.

Proof. Use Lemma 2.1.2 and (2.11) to check.

For the rest of this section, let Sndenote the group of permutations on ∆ ={1, 2, . . . , n}

and let

T :={s1, s2, . . . , sn−1}.

Recall that WT is the subgroup of W generated by{κs | s ∈ T }. In view of Lemma 2.7.1

we may make a definition.

Definition 2.7.2. Let ϵ : WT → Sn denote the homomorphism defined by

ϵ(G)(j) = Gj (1≤ j ≤ n) for G_{∈ W}T.

Lemma 2.7.3. ϵ : WT → Sn is an isomorphism.

Proof. It follows from that ∆ is a spanning set and that the n− 1 transpositions ϵ(κs1), ϵ(κs2), . . . , ϵ(κsn−1) generate Sn.

Proposition 2.7.4. The W-orbits of FS

2 are O0 ={0}, O1 ={a ∈ FS2 | a ̸= 0, ||a||s ≡ 1 or n − 2 (mod 4)}, (2.13) O2 ={a ∈ FS2 | a ̸= 0, ||a||s ≡ 2 or n − 3 (mod 4)}, O3 ={a ∈ FS2 | a ̸= 0, ||a||s ≡ 3 or n (mod 4)}, O4 ={a ∈ FS2 | a ̸= 0, ||a||s ≡ 0 or n − 1 (mod 4)}.

In particular O1 = O3 when n ≡ 1 (mod 4), O1 = O4 and O2 = O3 when n≡ 2 (mod 4),

Proof. It is clear that O0 is a W-orbit of FS2. There are four cases to put nonzero vectors

a, b in an orbit. (a) ||a||s = ||b||s : this is because ϵ(WT) = Sn acts transitively on the

fixed size subsets of ∆; (b)||b||s= n + 3− ||a||s or n− 1 − ||a||s: this is from (a) and the

observation that κsn||a||s=    n + 3− ||a||s if |∆(a) ∩ {n, n − 1, n − 2}| = 3, n− 1 − ||a||s if |∆(a) ∩ {n, n − 1, n − 2}| = 1, w(a) else (2.14)

by Lemma 2.7.1(ii) and (2.12); (c) ||a||s = ||b||s− 4 : this is by applying the first case

of (2.14) and then applying the second case of (2.14); and (d) _||a||s = ||b||s+ 4 : this is

by applying the second case of (2.14) and then the first case of (2.14). The proposition follows from the above cases (a)–(d).

For the rest of this section let Oi (0≤ i ≤ 4) denote the sets from Proposition 2.7.4.

Proposition 2.7.5. The representation κ is irreducible if and only if n is even.

Proof. Immediate from Proposition 2.7.4.

Corollary 2.7.6. We have |O1| =        2n−1− (−1)n42 n−2 2 if n≡ 0 (mod 4), 2n−1 _{if n≡ 1 (mod 4),} 2n−1_{+ (−1)}n−2_{4 2}n−2_{2 − 1 if n≡ 2 (mod 4),} 2n−2_{+ (}−1)n−3_{4 2}n−3_{2 if n}≡ 3 (mod 4). (2.15) Proof. By (2.13) we have |O1| =                        ∑ k≡1,2( mod 4) 1≤k≤n (_n k ) if n≡ 0 (mod 4), ∑ k≡1( mod 2) 1≤k≤n (_n k ) if n≡ 1 (mod 4), ∑ k≡0,1( mod 4) 1≤k≤n (_n k ) if n≡ 2 (mod 4), ∑ k≡1( mod 4) 1≤k≤n (_n k ) if n_{≡ 3 (mod 4),} where(n k )

is the binomial coefficient. From this we routinely prove (2.15) by induction on

n.

Let a _{∈ F}S

2. Recall that the isotropy group of a in W is {G ∈ W | Ga = a}. By the

elementary knowledge of group theory, the cardinality of the W-orbit of a is equal to the index of the isotropy group of a in W. For the rest of this section let

J :={s2, s3, . . . , sn}.

Observe that WJ is a subgroup of the isotropy group of es1 in W and that the W-orbit

of es1 is O1. Therefore |WJ||O1| divides |W|.

Proposition 2.7.7. Assume Γ is the Dynkin diagram of type E6. Then Ker κ = Z(W ).

2.7. Lit-only σ-game on the Dynkin diagram of type En

Proof. By Corollary 2.7.6 we have _|O1| = 27. By Lemma 2.6.2(ii) and Proposition 2.5.12

(the case D5), we know |WJ| = 245!. Since |WJ||O1| divides |W| we have |W| ≥ 27345.

By this and since|W | = 27₃4_{5 (for example see [21, p.44]), W is isomorphic to W and so}

Ker κ is trivial. By this and Corollary 2.3.2, Z(W ) is trivial.

In order to show Ker κ = Z(W ) in the cases E7 and E8, we cite [6, Lemma 10.2.11].

Lemma 2.7.8. ([6, Lemma 10.2.11]). Assume that Γ is one of simply-laced Dynkin

diagram of type E7 or E8. Then Z(W ) ={1, w◦}.

Proposition 2.7.9. Assume Γ is the Dynkin diagram of type E7. Then Ker κ = Z(W ).

Moreover Ker κ =_{{1, w◦}.}

Proof. By Corollary 2.3.2 and Lemma 2.7.8,|Kerκ| ≥ 2. By this and since |W | = 210₃4₅·7

(for example see [21, p.44]) we have|W| ≤ 29₃4_{5·7. By Corollary 2.7.6 we have |O}

1| = 28

and by Proposition 2.7.7 we have _|WJ| = 27345. Since |WJ||O1| divides |W| it follows

that |W| ≥ 29₃4₅· 7. Therefore |W| = 29₃4₅· 7 and this forces |Z(W )| = |Kerκ| = 2.

For the rest of this section we assume that Γ is the Dynkin diagram of type E8. Let

u_◦ denote the longest element of WJ.

Lemma 2.7.10. κ(u_◦)8 = 1 + 8.

Proof. By Lemma 2.7.8, u_◦ ∈ Z(WJ). Note that T ∩ J = {s2, s3, . . . , s7}, and that κ 

WT∩J is an isomorphism of WT∩J onto WT∩J by Lemma 2.6.2(ii) and Proposition 2.4.7.

Also ϵ WT∩J : WT∩J → S7 is an isomorphism , where ϵ is from Definition 2.7.2 and S7

is the group of permutations on{2, 3, . . . , 8}. Let

u′_◦ = κ−1(ϵ−1((2, 8, 3, 7, 4, 6, 5)))s8κ−1(ϵ−1((5, 8)(4, 7)(3, 6)))s8

κ−1(ϵ−1((4, 8)(3, 7)(2, 6)))s8κ−1(ϵ−1((5, 8)(4, 7)))s8

κ−1(ϵ−1((3, 7)(2, 6)))s8.

It is routine to check that the above u′_◦ maps to −I by the faithful representation defined in [11, p. 291] to conclude u′_◦ = u_◦. Therefore κ(u_◦) equals

ϵ−1((2, 8, 3, 7, 4, 6, 5))κs8ϵ −1_{((5, 8)(4, 7)(3, 6))κ} s8ϵ −1_{((4, 8)(3, 7)(2, 6))κ} s8 ϵ−1((5, 8)(4, 7))κs8ϵ −1_{((3, 7)(2, 6))κ} s8. (2.16) Applying (2.16) to 8 and using Lemma 2.7.1 and (2.12) for n = 8, the result follows.

Lemma 2.7.11. The restriction κ WJ of κ to J is injective.

Proof. Let κ′ denote the corresponding representation from WJ into GLJ(F2). From

Lemma 2.6.2(ii) and Proposition 2.7.7, we see that Ker κ  WJ ⊆ Kerκ′ = {1, u◦}.

By Lemma 2.7.10, u_◦ is not in Ker κ  WJ. Therefore Ker κ WJ is trivial and the result

follows.

We now can show Ker κ = Z(W ) in the case E8.

Proposition 2.7.12. Assume that Γ is the Dynkin diagram of type E8then Ker κ = Z(W ).

Moreover Ker κ =_{{1, w◦}.}

Proof. We have |O1| = 23 · 3 · 5 from Corollary 2.7.6 and |WJ| = |WJ| = 210345· 7

from Lemma 2.7.11. Note that _{|W | = 2}14₃5₅2_{7 (for example see [21, p.44]). It follows}

that _{|Kerκ| = 2. By Corollary 2.3.2 and Lemma 2.7.8, Kerκ and Z(W ) are equal to}

2.8 Summary

We now summarize the main results of this chapter.

Theorem 2.8.1. Let Γ denote a finite simple graph. Let W denote the Coxeter group of

type Γ. Let κ : W → GLS(F2) denote the representation from Theorem 2.2.2. Then the

following (i), (ii) are equivalent.

(i) Ker κ = Z(W ).

(ii) Γ is a simply-laced Dynkin diagram.

Proof. (i) ⇒ (ii): Recall that Z(W ) has finite order, from below Corollary 2.3.2. By

this and since W /Z(W ) ∼= W is finite, W has finite order. It is well-known that Γ is a simply-laced Dynkin diagram if and only if the Coxeter group W of type Γ is finite, for example see [21, p. 133]. Therefore (ii) follows.

(ii) _{⇒ (i): Immediate from Propositions 2.4.1, 2.4.8, 2.5.12, 2.7.7, 2.7.9, 2.7.12.}

Remark 2.8.2. Theorem 2.8.1 is probably known to some experts on Lie algebras [3, 4,

5, 22].

simply-laced Dynkin diagrams reducibility of κ Ker κ

An (n≥ 1) κ isirr. iff n = 1 or n is even.

{ {1, w◦} if n = 1, {1} else. Dn (n≥ 4) κ is not irr. { {1, w◦} if n is even, {1} else. E6 κ is irr. {1} E7 κ is not irr. {1, w◦} E8 κ is irr. {1, w◦}

2.8. Summary Γ W-orbits ofFS 2 An (n≥ 1) Oi ={a ∈ FS2 | ||a||s= i or n + 1− i} (0 ≤ i ≤ ⌊n+1₂ ⌋). Dn (n≥ 4) Oi ={a ∈ Z | ||a||s= i or n− i} (0 ≤ i ≤ ⌊n₂⌋), Ωo ={a ∈ FS2 \ Z | ||a||s≡ 1 or n − 1 (mod 2)}, Ωe ={a ∈ FS2 \ Z | ||a||s≡ 0 or n (mod 2)}, Ωo = Ωe=FS2 \ Z when n is odd. En (n≥ 6) O0={0}, O1={a ∈ FS2 | a ̸= 0, ||a||s≡ 1 or n − 2 (mod 4)}, O2={a ∈ FS2 | a ̸= 0, ||a||s≡ 2 or n − 3 (mod 4)}, O3={a ∈ FS2 | a ̸= 0, ||a||s≡ 3 or n (mod 4)}, O4={a ∈ FS2 | a ̸= 0, ||a||s≡ 0 or n − 1 (mod 4)}. O1= O3 when n≡ 1 (mod 4),

O1= O4 and O2= O3 when n≡ 2 (mod 4), O2= O4 when n≡ 3 (mod 4),

O1= O2 and O3= O4 when n≡ 0 (mod 4).

Table 2: the W-orbits of_FS

Chapter 3

Lit-only sigma-game on a graph with

a long induced path

For a∈ FS

2 let||a|| denote the number of on vertices of Γ that are assigned by a, and

we call _{||a|| the weight of a. For a subset O of F}S

2 define||O|| to be

min

a∈O||a||.

Motivated by a goal of lit-only σ-game, we consider the following numbers.

Definition 3.0.3. Let k _{≥ 1 denote an integer. We say that Γ is k-lit for lit-only σ-game}

whenever ||O|| ≤ k for any W -orbit O of FS

2.

Definition 3.0.4. ([26]) Let µ(Γ) denote the minimum number k such that Γ is k-lit for

lit-only σ-game. We call µ(Γ) the minimum light number for lit-only σ-game on Γ. There are three known results about µ(Γ). If Γ is a simply-laced Dynkin diagram then

µ(Γ) = 1 (see [2] or [8]). If Γ is the graph En (n ≥ 6) shown in Figure 1.2 then one can

use Proposition 2.7.4 to check µ(Γ) = 1. If Γ is a tree with ℓ leaves X. Wang and Y. Wu [26] prove µ(Γ)≤ ⌈ℓ/2⌉. In this chapter we consider an extension of simply-laced Dynkin diagrams: an n-vertex graph with an induced path of n− 1 vertices. In Chapter 2 we studied the lit-only σ-game on a simply-laced Dynkin diagram with the help of a specific basis forFS

2. We extend the idea to this case. We shall find a criterion of µ(Γ) and give a

description of W-orbits ofFS

2 for this case.

For the rest of this chapter we adopt the following assumption.

Assumption 3.0.5. Assume that Γ = (S, R) is a simple connected graph whose vertex

set S = {s1, s2, . . . , sn} (n ≥ 2). Suppose the sequence s1, s2, . . . , sn−1 forms an induced

path in Γ. Let j1, j2, . . . , jm (m ≥ 1) denote a subsequence of 1, 2, . . . , n − 1 such that

c c c c c c c c c c sn sn−1 sn−2 sn−3 sjm sj2 sj1 s3 s2 s1 q q q q q q q q q q q q q q q q q q

Figure 2.1: an n-vertex graph with an induced path of n− 1 vertices.

3.1

The sets Π, Π

and Π

In this chapter let

1 = es1, i + 1 = κsiκsi−1· · · κs11 (1≤ i ≤ n − 1), n + 1 = esn. (3.1) Let Π ={1, 2, . . . , n}, (3.2) Π0 ={i ∈ Π | i t n + 1 = 0}, (3.3) Π1 = Π\ Π0. (3.4)

For convenience let es0 = 0. From (3.1) and the construction,

Π0 ={i | i = esi−1 + esi, 1≤ i ≤ n − 1 or i = esn−1},

Π1 ={i | i = esi−1 + esi + esn, 1≤ i ≤ n − 1 or i = esn−1 + esn}.

Note that 1 _{≤ |Π}0|, |Π1| ≤ n − 1 and |Π0| + |Π1| = n. For convenience let jm+1 = n and

jm+2 = n. Observe that

Π0 ={i ∈ Π | i ∈ (0, j1]∪ (j2, j3]∪ · · · ∪ (j2k, j2k+1]}, (3.5)

Π1 ={i ∈ Π | i ∈ (j1, j2]∪ (j3, j4]∪ · · · ∪ (j2k−1, j2k]}, (3.6)

where k = ⌈m

2⌉ and (a, b] = {x | x ∈ Z, a < x ≤ b}. We now establish some lemmas for

later use. Proposition 3.1.1. |Π1| = ⌈m 2⌉ ∑ k=1 j2k− j2k−1.

Proof. Immediate from (3.6).

For the rest of this chapter let

3.1. The sets Π, Π0 and Π1

Lemma 3.1.2. For 1_{≤ i ≤ n − 1 we have}

1 + 2 +· · · + i = { esi+ esn if |[i] ∩ Π1| is odd, esi if |[i] ∩ Π1| is even, and 1 + 2 +· · · + n = { esn if |Π1| is odd, 0 if |Π1| is even. Proof. Use (3.1). Lemma 3.1.3. ∑ i∈Π0 i = m ∑ k=1 es_jk.

Proof. Use Lemma 3.1.2 and (3.5) to verify this.

Lemma 3.1.4. κsii = i + 1, κsii + 1 = i and κsi fixes other vectors in Π\ {i, i + 1} for

1≤ i ≤ n − 1.

Proof. Immediate from (3.1).

For the rest of this chapter let Sndenote the symmetric group on Π. From Lemma 3.1.4,

κsi acts on Π as the transposition (i, i + 1) in Sn for 1≤ i ≤ n − 1.

Corollary 3.1.5. Let U denote the subspace of FS

2 spanned by the vectors in Π. Then U

is closed under W.

Proof. By Lemma 3.1.4, U is closed under the action of κs1, κs2, . . . , κsn−1. For i∈ Π we

have κsni =    i if i∈ Π0, i +∑ j∈Π0 j if i_{∈ Π}1

by Lemma 3.1.3. It follows that κsni lies in U. The result follows.

For the rest of this chapter let U denote the subspace of FS

2 from Corollary 3.1.5.

Proposition 3.1.6. If |Π1| is odd then Π is a basis for U; if |Π1| is even then for any

j ∈ Π, Π \ {j} is a basis for U. Moreover esn ̸∈ U if |Π1| is even.

Proof. By Lemma 3.1.2, 1, 2, . . . , n_{− 1 are linearly independent and hence U has}

dimen-sion at least n− 1. Since esn ̸∈ Span{1, 2, . . . , n − 1}, the proposition follows from the

second case of Lemma 3.1.2.

For the rest of this chapter let P denote the subset of S consisting of s1, s2, . . . , sn−1.

Recall that WP denotes the subgroup of W generated by κs1, κs2, . . . , κsn−1.

Corollary 3.1.7. The subgroup WP of W is isomorphic to the symmetric group Sn on

Π.

3.2 The simple basis ∆ of

F

2 To better describe the W-orbits of _FS

2 we choose a specific basis of FS2. Let

∆ := {

Π if |Π1| is odd,

Π∪ {n + 1} \ {n} if _|Π1| is even.

By Proposition 3.1.6, ∆ is a basis of_FS

2. We call ∆ the simple basis ofFS2. For each u∈ FS2,

u can be uniquely written as a linear combination of elements in ∆, so let ∆(u) denote

the subset of ∆ such that

u = ∑

i∈∆(u)

i.

Let||u||s:=|∆(u)|. We refer to ||u||sas the simple weight of u. Note that for 1≤ i ≤ n−1,

the vector 1 + 2 +_{· · · + i has simple weight i but has weight}

||1 + 2 + · · · + i|| =

{

1 if |[i] ∩ Π1| is even,

2 if |[i] ∩ Π1| is odd

by Lemma 3.1.2.

In the next two sections we shall give a description of W-orbits of _FS

2. For convenience

we adopt the following notation. For V ⊆ FS

2 and T ⊆ {0, 1, . . . , n} define

VT :={u ∈ V | ||u||s∈ T }.

For shortness Vt1,t2,...,ti := V{t1,t2,...,ti} where t1, t2, . . . , ti ∈ {0, 1, . . . , n}. Let odd denote the

set of all odd integers among _{{0, 1, . . . , n}.}

3.3

The case

|Π

| is odd

In this section we assume |Π1| to be odd and the counter part is treated in the next

section. In this case U = _FS

2 and so ∆ ={1, 2, . . . , n} is a basis of FS2. By Lemma 3.1.2

we have esi = { 1 + 2 +· · · + i if _{|[i] ∩ Π}1| is even, i + 1 + i + 2 +· · · + n if _{|[i] ∩ Π}1| is odd, (1≤ i ≤ n − 1), and esn = 1 + 2 +· · · + n. Hence we have ||esi||s= { i if |[i] ∩ Π1| is even, n− i if _{|[i] ∩ Π}1| is odd, (1≤ i ≤ n − 1)

and ||esn||s = n. Therefore there exists a vector with simple weight i and weight 1 if

and only if |[i] ∩ Π1| is even, i = n or |[n − i] ∩ Π1| is odd. Let [i] := {1, 2, . . . , i} for

i = 1, 2, . . . , n. Let

K :={i ∈ [n] | |[i] ∩ Π1| is even, i = n or |[n − i] ∩ Π1| is odd}. (3.7)

By Lemma 3.1.2, ||Ui|| ≤ 2 for 1 ≤ i ≤ n. Note that

3.3. The case |Π1| is odd Lemma 3.3.1. For u_{∈ F}S 2 we have κsnu = { u if _{|∆(u) ∩ Π}1| is even, u + ∑ i∈Π0 i else. Moreover ||κsnu||s = {

||u||s if |∆(u) ∩ Π1| is even,

n + 2k− |Π1| − ||u||s else,

where k =|Π1∩ ∆(u)|.

Proof. If_{|∆(u) ∩ Π}1| is even then utesn = 0 and κsnu = u by construction. If |∆(u) ∩ Π1|

is odd, then κsnu = u + m ∑ k=1 es_jk = u +∑ i∈Π0 i

by Lemma 3.1.3, and||κsnu||s =|∆(u)∩Π1|+ (|Π0|− |∆(u)∩Π0|) = n+2k −|Π1|− ||u||s.

The result follows.

Lemma 3.3.2. The WP-orbits of FS2 are {0} and Ui for 1≤ i ≤ n.

Proof. Immediate from Corollary 3.1.7 and ∆ = Π.

We now give a description of W-orbits of FS

2 and characterize µ(Γ) in the case 3 ≤

|Π1| ≤ n − 3.

Theorem 3.3.3. Assume that 3≤ |Π1| ≤ n − 3. Then the W-orbits of FS2 are {0}, UA1, UA2, UA3, UA4, where

Ai :={j ∈ [n] | j ≡ i, n + |Π1| − i (mod 4)}.

Moreover the number of W-orbits of FS

2 is 3 if n is even and 4 if n is odd.

Proof. Fix an integer 1 _{≤ i ≤ n. By Lemma 3.3.2, U}i is a W-orbit of FS2. Note that

W is the subgroup of GLS(F2) generated by WP and κsn. By the above comments and

by Lemma 3.3.1, the union of those Ui,n+2k−|Π1|−i forms a W-orbit of F

S

2, where k runs

through possible odd integers _|Π1 ∩ ∆(u)| for u ∈ Ui. In fact k is any odd number that

satisfies k≤ |Π1| and 0 ≤ i − k ≤ |Π0|; equivalently

max{1, i + |Π1| − n} ≤ k ≤ min{|Π1|, i}. (3.9)

Such an odd integer k exists for any 1≤ i ≤ n, and note that

n + 2k− |Π1| − i ≡ n + |Π1| − i (mod 4)

since k and _|Π1| are odd integers. To see the W-orbits of FS2 as stated in the theorem, it

the least odd integer greater than or equal to max_{{1, i + |Π}1| − n + 2}. For this k, (3.9)

holds and then Ui,n+2k−|Π1|−iis contained in a W-orbit ofF

S

2. Here we use the assumption

|Π1| ≤ n−3 to guarantee the existence of such k. Replacing (i, k) by (n+2k−|Π1|−i, k+2)

in (3.9) we have

max{1, 2k − i} ≤ k + 2 ≤ min{|Π1|, n + 2k − |Π1| − i}. (3.10)

The above k and the assumption 3≤ |Π1| guarantee the equation (3.10). Since n + 2(k +

2)− |Π1| − (n + 2k − |Π1| − i) = i + 4 we have Ui+4,n+2k−|Π1|−i is contained in a W-orbit

of _FS

2. Putting these together, Ui,i+4 is in a W-orbit of FS2. The result follows.

Corollary 3.3.4. Assume that 3≤ |Π1| ≤ n − 3. Then

µ(Γ) =

{

1 if Ai∩ K ̸= ∅ for all i,

2 else,

where K is defined as (3.7).

Proof. Use (3.8) and Theorem 3.3.3.

We now consider the cases _|Π1| = 1, n − 2, n − 1.

Theorem 3.3.5. Assume that |Π1| = 1, n − 2 or n − 1. Then the W-orbits of FS2 are {0}

and _   Ui,n+1−i if |Π1| = 1, Uodd, U2j if |Π1| = n − 2, U2i−1,2i if |Π1| = n − 1 for 1≤ i ≤ ⌈n

2⌉ and 1 ≤ j ≤ (n − 1)/2. Moreover the number of W-orbits of F

S 2 is    ⌈(n + 2)/2⌉ if _|Π1| = 1, (n + 3)/2 if |Π1| = n − 2, (n + 2)/2 if |Π1| = n − 1.

Proof. As the proof in Theorem 3.3.3, Ui,n+2k−|Π1|−iis contained in a W-orbit ofF

S

2, where

k needs to satisfy (3.9). In the case |Π1| = 1, k = 1 is the only possible choice and hence

Ui,n+1−i is a W-orbit of FS2. In the case |Π1| = n − 2, we have k = i − 2 or i if i is odd;

k = i− 1 if i is even. In the case |Π1| = n − 1, we have k = i if i is odd; k = i − 1 if i is

even. In each of the remaining the proof follows similarly.

Corollary 3.3.6. Assume that _|Π1| = 1, n − 2 or n − 1. Then µ(Γ) ≤ 2. Moreover

µ(Γ) = 1 if and only if    {i, n + 1 − i} ∩ K ̸= ∅ for 1≤ i ≤ ⌈n 2⌉ if |Π1| = 1, odd∩ K ̸= ∅, U2j ∩ K ̸= ∅ for 1≤ j ≤ ⌊n₂⌋ if |Π1| = n − 2,

{2i − 1, 2i} ∩ K ̸= ∅ for 1_{≤ i ≤ ⌈}n

2⌉ if |Π1| = n − 1,

where K is defined as (3.7).

Proof. Use (3.8) and Theorem 3.3.5.

We end this section with an example.

Example 3.3.7. Let Γ be an odd cycle of length n; i.e. n is odd, m = 2, j1 = 1 and

j2 = n− 1. Then Π0 ={1, n} and Π1 ={2, 3, . . . , n − 1}. Note that |Π1| = n − 2 is odd,

and K =_{{1, 3, . . . , n}. By Theorem 3.3.5 we have the W-orbits of F}S

2 are

{0}, Uodd, U0, U2, U4, . . . , Un−1.

3.4. The case|Π1| is even

3.4

The case

|Π

| is even

In this section we assume that |Π1| is even. In this case ∆ = Π ∪ {n + 1} \ {n} is a

basis for _FS

2 and ∆\ {n + 1} is a basis for U. Recall that

1 + 2 +· · · + n = 0 (3.11)

Let U :=FS

2 \ U. Note that Un =∅, U = n + 1 + U and U1 ={n + 1}. By Lemma 3.1.2

we have esi = { 1 + 2 +· · · + i ∈ U if _{|[i] ∩ Π}1| is even, 1 + 2 +· · · + i + n + 1 ∈ U if |[i] ∩ Π1| is odd, (1≤ i ≤ n − 1), and esn = n + 1∈ U. It follows that ||esi||s = { i if _{|[i] ∩ Π}1| is even, i + 1 if |[i] ∩ Π1| is odd, (1≤ i ≤ n − 1),

and ||esn||s = 1. Therefore there exists a vector in U with simple weight i and weight 1 if

and only if|[i] ∩ Π1| is even; there exists a vector in U with simple weight i and weight 1

if and only if _{|[i − 1] ∩ Π}1| is odd or i = 1. For the rest of this section let

K :={i ∈ [n − 1] | |[i] ∩ Π1| is even}, (3.12)

L := {i ∈ [n] | |[i − 1] ∩ Π1| is odd or i = 1}. (3.13)

Note that||Ui||, ||Uj|| ≤ 2 and that

||Ui|| = 1 if and only if i∈ K,

||Uj|| = 1 if and only if j ∈ L

for 1≤ i ≤ n − 1 and 1 ≤ j ≤ n.

Lemma 3.4.1. For u_{∈ F}S

2 let k =|Π1∩ ∆(u)|. Then the following (i), (ii) hold.

(i) For u∈ U we have

κsnu = { u if |∆(u) ∩ Π1| is even, u + ∑ i∈Π0 i else. Moreover ||κsnu||s=   

||u||s if |∆(u) ∩ Π1| is even,

n + 2k− |Π1| − ||u||s if |∆(u) ∩ Π1| is odd and n ∈ Π1,

(ii) For u_{∈ U we have} κsnu = { u if |∆(u) ∩ Π1| is odd, u + ∑ i∈Π0 i else. Moreover ||κsnu||s =   

||u||s if |∆(u) ∩ Π1| is odd,

n + 2k + 2− |Π1| − ||u||s if |∆(u) ∩ Π1| is even and n ∈ Π1,

||u||s+|Π1| − 2k else.

Proof. The proof is similar to the proof of Lemma 3.3.1, except that at this time since

the choice of simple basis ∆ is different, the action of κsn on a vector is a little different,

and we need to use (3.11) to adjust the simple weight of a vector.

In view of Corollary 3.1.5 we discuss the W-orbits (resp. WP-orbits) of FS2 into the

two parts, one in U and the other in U .

Lemma 3.4.2. The WP-orbits of FS2 are{0}, U1, Ui+1,n+1−i and Ui,n−i for 1≤ i ≤ ⌊n₂⌋.

Proof. By construction U1 = {esn} is a WP-orbit of F

S

2. By Corollary 3.1.5 and

Corol-lary 3.1.7, Ui is contained in a WP-orbit of U and Ui+1 is in a WP-orbit of U for

1 ≤ i ≤ n − 1. By (3.11), Ui,n−i is contained in a WP-orbit of FS2 and Ui+1,n+1−i is

in a WP-orbit of U for 1 ≤ i ≤ n − 1. Since no other ways to put these sets together the

result follows.

Theorem 3.4.3. Assume that 4≤ |Π1| ≤ n − 3. Then the W-orbits of FS2 are {0}, UB1, UB2, UB3, UB4, UC1, UC2, UC3, UC4, where

Bi ={j ∈ [n − 1] | j ≡ i, i + |Π1| − 2, n − i, n − i + |Π1| − 2 (mod 4)},

Ci ={j ∈ [n] | j ≡ i, i + |Π1|, n + 2 − i, n + 2 − i + |Π1| (mod 4)}.

Moreover the number of W-orbits of _FS

2 is 6 if n is even and 4 if n is odd.

Proof. We first determine the W-orbits of U. By Lemma 3.4.2, Ui,n−i is contained in a

W-orbit of U for 1 ≤ i ≤ n − 1. Suppose n ∈ Π0 and the case n ∈ Π1 is left to the

reader. In this case Ui,i+|Π1|−2k is contained in a W-orbit of U by Lemma 3.4.1(i), where

1 ≤ i + |Π1| − 2k ≤ n − 1 and k runs through possible odd integers |Π1 ∩ ∆(u)| for

u∈ Ui. In fact k is any odd number that satisfies k ≤ |Π1| − 1 and 0 ≤ i − k ≤ |Π0| − 1;

equivalently

max{1, i + |Π1| − n + 1} ≤ k ≤ min{|Π1| − 1, i}. (3.14)

Such an odd k exists for any 1_{≤ i ≤ n − 3, and note that}

i +|Π1| − 2k ≡ i + |Π1| − 2 (mod 4).

To determine the W-orbits of U, it remains to show that Ui,i+4 is contained in a W-orbit

of U for 1 _{≤ i ≤ ⌊}n

2⌋. Suppose 4 ≤ |Π1| ≤ 6. Set k = 1 to conclude that Ui,i+2 in a

W-orbit of U if _|Π1| = 4; Ui,i+4 in a W-orbit of U if |Π1| = 6. Thus we suppose that

3.4. The case|Π1| is even

or equal to max_{{1, i + |Π}1| − n + 3}. For this k, (3.14) holds and then Ui,i+|Π1|−2k is

contained in a W-orbit of U . Here we use the assumption |Π1| ≤ n − 3. Replacing (i, k)

by (i +|Π1| − 2k, |Π1| − k − 2) in (3.14) we have

max_{{1, i + 2|Π}1| − 2k − n + 1} ≤ |Π1| − k − 2 ≤ min{|Π1| − 1, i + |Π1| − 2k}. (3.15)

The above k, the assumption 4≤ |Π1| and i ≤ n − 6 guarantee the equation (3.15). Since

(i +|Π1| − 2k) + |Π1| − 2(|Π1| − k − 2) = i + 4 we have Ui+4,i+|Π1|−2k in a W-orbit of U .

Putting these together, Ui,i+4 is contained in a W-orbit of U. Therefore the W-orbits of

U are UB1, UB2, UB3, UB4.

We next determine the W-orbits of U . Since the proof is similar to the above case, we only give a sketch. By Lemma 3.4.2, Ui,n+2−i is contained in a W-orbit of U for 2≤ i ≤ n.

We suppose n∈ Π1 and leave the case n∈ Π0 to the reader. By Lemma 3.4.1(ii) we have

Ui,n+2k+2−|Π1|−i is contained in a W-orbit of U , where k =|∆(u) ∩ Π1| is an even number

for some u_{∈ U}i and 1≤ i ≤ n − 4. By the same argument with replacing k by k + 2 we

find Ui+4,n+2k+2−|Π1|−i is contained in a W-orbit of U . Therefore Ui,i+4 is contained in a

W-orbit of U . We have determined the W-orbits of FS

2. The result follows.

Corollary 3.4.4. Assume that 4≤ |Π1| ≤ n − 3. Then

µ(Γ) =

{

1 if Bi∩ K ̸= ∅ and Ci∩ L ̸= ∅ for all i,

2 else,

where K and L are defined as (3.12) and (3.13), respectively. Proof. Use (3.12), (3.13) and Theorem 3.4.3.

We now consider the cases |Π1| = 2, n − 2, n − 1.

Theorem 3.4.5. Assume that |Π1| = 2, n − 2 or n − 1. Let the sets C1, C2 be as in

Theorem 3.4.3. Then the W-orbits of FS

2 are {0} and    Ui,n−i, UC1, UC2 if |Π1| = 2, Uodd, U2j,n−2j, Uodd, U2t,n+2−2t if |Π1| = n − 2, U2j−1,2j,n−2j,n+1−2j, U2t−1,2t,n+2−2t,n+3−2t, if |Π1| = n − 1 for 1≤ i ≤ ⌊n 2⌋, 1 ≤ j ≤ ⌈ n−2 4 ⌉ and 1 ≤ t ≤ ⌈ n

4⌉. Moreover the number of W-orbits of F

S

2

is _{

(n + 6)/2 if _|Π1| = 2 and n is even, or |Π1| = n − 2,

(n + 3)/2 if _|Π1| = 2 and n is odd, or |Π1| = n − 1.

Proof. The proof is similar to the proof of Theorem 3.3.5 that follows from the proof of

Theorem 3.3.3. At this time, to determine the W-orbits of U we check what values of odd k occur in (3.14) in each case of|Π1| ∈ {2, n − 2, n − 1}. To determine the W-orbits

of U , we do similarly as in the second part of the proof of Theorem 3.4.3.

Corollary 3.4.6. Assume that _|Π1| = 2, n − 2 or n − 1. Then µ(Γ) ≤ 2. Moreover

µ(Γ) = 1 if and only if {i, n − i} ∩ K ̸= ∅ C1∩ L ̸= ∅, C2∩ L ̸= ∅ for 1≤ i ≤ ⌊n₂⌋ if |Π1| = 2, { odd∩ K ̸= ∅, {2j, n − 2j} ∩ K ̸= ∅ for 1≤ j ≤ ⌈n−2 4 ⌉ odd∩ L ̸= ∅, {2t, n + 2 − 2t} ∩ L ̸= ∅ for 1_{≤ t ≤ ⌈}n 4⌉ if |Π1| = n − 2, { {2j − 1, 2j, n − 2j, n + 1 − 2j} ∩ K ̸= ∅ for 1≤ j ≤ ⌈n−2 4 ⌉ {2t − 1, 2t, n + 2 − 2t, n + 3 − 2t} ∩ L ̸= ∅ for 1_{≤ t ≤ ⌈}n 4⌉ if |Π1| = n − 1,