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Representations of sln(C), the
Weyl Group, the Root Lattice,
and Linear Braids
Kie Seng Nge
Supervisor: Assoc. Prof. Anthony Licata
October, 2017
A thesis submitted for the degree of Bachelor of Science (Advanced) (Honours)
of the Australian National University
Dedicated to my supportive parents.
Declaration
Except where otherwise stated, this thesis is my own work prepared under the supervision
of Assoc. Prof. Anthony Licata.
Kie Seng Nge
Acknowledgements
First and foremost, I would like to give thanks to the almighty God for the strength
and wisdom He has given throughout my completion of this papaer.
To my supervisor, Anthony Licata, I would like to express my heartfelt thanks for
the opportunity to work with him on my Honours thesis. His unceasing guidance and
insight in his research area have kept me motivated throughout the learning process.
The inspiration and knowledge from him has been extremely invaluable. Not forgetting,
thanks to David Symth who assisted me in the relevant background reading during
Anthony’s absence.
Besides that, I would like take this opportunity to thank our Honours convenor, Joan
Licata for coordinating the Honours program and our Honours conference coordinator,
Bryan Wang for organising the mandatory Honours conference. I also want to thank
Arnab Saha who is always willing to share his knowledge on number theory and his
career experience with me, helping me to map out my future path.
In addition, I would like to express my utmost gratitude to all lecturers who have
mentored me throughout my pursuit of Bachelor of Science (Advanced) (Honours) in
Pure Mathematics. They are John Urbas, Vigleik Angeltveit, Jim Borger, Amnon
Neeman, Scott Morrison, Brett Parker, Jesse Burke, Bregje Pauwels, Ben Andrews,
Qi-Rui Li, Martin Ward, Michael Norrish, Tim Trudgian, Pierre Portal, Vladimir
Mangazeev, Dayal Wickramasinghe, Griffith Ware, Peter Bouwknegt and Linda Stals.
Many thanks to a postdoctoral fellow, Changwei Xiong and a former Honours
student, Suo Jun Tan whom I had consultation with whenever I had difficulty with
the study content. Furthermore, I would like to thank my best friend, James Bailie, as
well as other peers – An Ran Chen, Edmund Heng, and Joanne Zheng for constantly
exchanging ideas with me keeping every semesters enjoyable. There are also times when
we struggled, but these memories are unique and will last forever.
Apart from the above, my family deserve a special mention. They are my powerful
vii
backing; they are always on my side whenever I am up or down. Without their
unwavering support, I might not have made it this far in both academics and life in
general. Also, thank you to my family members in Christ for their moral and spiritual
support.
Last but not least, I thank my beloved country, Malaysia for recognizing my abilities
and accepting me as a MybrainSc scholar, which has rendered me a burden-free academic
life here by settling my expenses and tuition fees at the Australian National University
(ANU).
viii
Contents
Acknowledgements vii
Introduction 1
1 Representations of the Special Linear Lie Algebra sln(C) 5
1.1 What is a Representation of sln(C)? . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 Semisimplicity of sln(C) . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 General theory of semisimple Lie algebra . . . . . . . . . . . . . 11
1.2 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 Cartan decomposition of sln(C) . . . . . . . . . . . . . . . . . . . 13
1.2.2 Representation of sl2(C) . . . . . . . . . . . . . . . . . . . . . . 14
1.2.3 The highest weight vector in a representation . . . . . . . . . . . 17
1.3 Full Picture of a Representation . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 Finding all weights . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.2 Multiplicity of the weights in an irreducible representation . . . . 23
1.4 Classifying the Finite Dimensional Irreducible Representations . . . . . 26
2 Weyl Group Acting on the Root Lattice 29
2.1 Root System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.1 Killing form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.2 Properties of roots . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Reflections in the Root Lattice . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Definition of a Weyl group . . . . . . . . . . . . . . . . . . . . . 36
2.2.2 An isomorphism between W(sln(C)) and Sn . . . . . . . . . . . . 37
2.3 An example – S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
ix
3 Combinatorics in the Weyl Group of sln(C) 43
3.1 Coxeter System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 A bijection between reflections and roots . . . . . . . . . . . . . 45
3.1.3 Minimal length expressions . . . . . . . . . . . . . . . . . . . . . 49
3.2 Expressions in Weyl Group of sln(C) . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Reduced expression in W(sln(C)) . . . . . . . . . . . . . . . . . . 53
3.2.2 The longest element in Weyl group of sln(C) . . . . . . . . . . . 56
3.3 Root Sets for Expressions in the Weyl Group . . . . . . . . . . . . . . . 57
4 From the Weyl Group of sln(C) to the Braid Group Bn 61
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Positive and Negative Root Sets for Elements of the Braid Group . . . . 63
4.3 Separated Root Sets and Linear Braids . . . . . . . . . . . . . . . . . . . 65
5 Presentations for the Braid Group and their Length Functions 75
5.1 The Abstract Group, Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 The Abstract Group, Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Relation between `linear and `Garside . . . . . . . . . . . . . . . . . . . . 80
Bibliography 82
x
Introduction
Chapter 1 is devoted to studying the finite dimensional representation of the semisimple
Lie algebra sln(C) systematically. First, we define the representation of sln(C). Next,
we study the case of sl2(C) thoroughly. The finite dimensional irreducible representa-
tions of sl2(C) act as building blocks for the representation of sln(C). Subsequently,
we prove the most important theorem (Theorem 1.4.1) in this chapter; this theorem
classifies the representations of sln(C) with their highest weight. With the knowledge
of the highest weight of the representation, we can reveal all weights contained in the
representation and also their corresponding multiplicity. While exploring the repre-
sentation of sln(C), we discover an interesting symmetry group, called Weyl group W,
acting on the root lattice generated by the weights.
In Chapter 2, we study the symmetry group in the root lattice of sln(C). The Weyl
group is generated by the reflections in root lattice. We also spend a substantial effort
in showing the set of roots in the adjoint representation of sln(C) forms a root system.
A mechanics – Killing form is introduced to tackle this. Killing form of sln(C) put
forward an idea of length and angle in the root lattice. The four key features of roots
in Theorem 2.1.12 are what constitute an abstract root system. We then finish this
chapter by proving the isomorphism between the Weyl group of sln(C) and the sym-
metric group Sn (Theorem 2.2.10). This can be easily observed by suitably labelling
the roots. A relevant example S4 is provided to demonstrate the claim.
In Chapter 3, we begin to explore the combinatorics in the Weyl group W ∼= Sn of
sln(C). With the identification of the Weyl group with the symmetric group in Chapter
2, the Weyl group W of sln(C) possesses the properties that hold in a more general
group, namely the Coxeter group. The symmetric group Sn is one of the classic exam-
ple. We describe the characteristics of elements in W(sln(C)) in the Coxeter language.
The most prominent theorem in this chapter is to relate combinatorics of words to the
action of W(sln(C)) on the root lattice (Corollary 3.2.3). Before the chapter ends, we
derive a few traits on the left and right root set of reduced expressions.
So far, there are no new knowledge being tabled in the first three chapters. Our
original works start in Chapter 4.
1
In Chapter 4, we first introduce linear braids, which are certain distinguished lifts
from the Weyl group of sln(C) to the braid group, determined by a splitting of the root
set of the Weyl group element (Definition 4.3.9). In particular, we define a notion of
separated root sets (Definition 4.3.1). In a pair of separated root sets, the non-negative
cones defined by the root sets intersect only at the origin. Geometrically, we can use a
hyperplane to divide the two separated root sets with each of them lying opposite side
of the hyperplane.
The notion of a linear braid comes from braid group actions on categories. However,
this thesis will not discuss the action of the braid group on categories. It is merely to
motivate our topic. There is a special categorical action of the braid group on the
homotopy category of projective modules over the zigzag algebra. More precisely, to
each braid β ∈ Bn, there is an object
F (β) ∈ Kb(An − bimod).
The algebra An is a finite-dimensional algebra called the zigzag algebra, and the ob-
jects of Kb(An − bimod) are chain complexes of (An, An)-bimodules, considered up to
homotopy. This is, in fact, a categorical analogue of the notion of a representation of
Bn on a vector space.
Without defining these notions here, the category Kb(An−bimod) is actually trian-
gulated with a canonical t-structure. As a consequence, the category Kb(An − bimod)
has a canonically-defined abelian subcategory - the heart of the canonical t-structure -
which we denote by C. So, one can question which braids β have the property that the
object F (β) lives in the heart C. The answer is the following:
Theorem 0.0.1 (Licata). The object F (β) lives in the heart of the t-structure if and
only if β is β is linear.
Moreover, Licata conjectures an alternative combinatorial characterisation of linear
braids. The following is what he has conjectured:
Conjecture 0.0.2 (Licata). The braid β is linear as in Definition 4.3.9 if and only if
β is the product of a positive Garside generator and a negative Garside generator as
defined in Definition 4.1.9.
We prove Conjecture 0.0.2 as Theorem 4.3.25 in this thesis. This is the main theo-
rem we have achieved when studying the linear braids.
Finally, in Chapter 5, there are three natural collections of generators for the braid
group to be considered. They are the Artin generators, the Garside generators, and the
2
linear generators, which are linear braids discovered in the previous chapter. The main
things we did are proposing two new presentations of the braid group with respect to the
Garside generators and the linear generators (Theorem 5.1.1 and Theorem 5.2.1). After
that, we analyse the length functions with respect to these generators by pinpointing
the nexus between the length function in the Garside generators and the length function
in the linear generators.
3
4
Chapter 1
Representations of the Special
Linear Lie Algebra sln(C)
In the mid nineteenth century, the mathematical object, Lie group arose when a Nor-
wegian mathematician, Sopus Marius Lie(1842 -1849) started to explore every possible
group actions on manifolds locally. Lie groups are smooth manifolds studied signif-
icantly in differential geometry. Lie algebra, named by Hermann Weyl after Sophus
Lie in the 1930s, is simply an algebraic object dismissing the topological complexity
of Lie group. Lie group’s algebraic structure embraces most of the topology of the
groups, although it seems at first we drop too much of its details. From elementary
Lie algebra theory, semisimple Lie algebras are sometimes regarded as atomic objects.
Their representations are also interesting ones. In this thesis, we will focus first on the
representations of the complex semisimple Lie algebra, sln(C).
In Section 1.1, we introduce the classic object – sln(C) in the representation theory.
It is a complex semisimple Lie algebra which is simply a vector field with some extra
properties. Moreover, we study a few representations of sln(C) by looking at how they
act as a linear map on a fixed vector space. We also show that sln(C) is semisimple
from the definition of semisimple concerning ideals in a Lie algebra. Its semisimplicity
allows the Jordan decomposition to be preserved under any representation. This is
extremely important when we are studying representations of sln(C).
In Section 1.2, we analyse a far more distinguishable representation of sln(C) – its
adjoint representation. However, to do this, we must first start off with the irreducible
finite dimensional representation of sl2(C). The reason is that it is the building block
of the adjoint representation of sln(C) for n > 2. From here, we are able to generalize a
couple of observations in the representation of sl2(C) to that of sln(C) (Theorem 1.2.22).
This section is also the first place we introduce the idea of a root space.
5
In Section 1.3, having found a highest weight vector v in a finite-dimensional rep-
resentation of sln(C), we wish to obtain the rest of its weight vectors. This is possible
since we are only considering finite dimensional representations. In the searching pro-
cess, we discover that the weights actually form a lattice where an isometry group
generated by reflections acts on. The Weyl group is first tracked down when it is used
to fill the weight diagram of a finite dimensional representation. However, the number
of distinct weights and the dimension of the representation do not match up. In truth,
some weights have multiplicity more than one. Therefore, we use Young tableau to
assist us to find the multiplicity of weights in an irreducible representation.
Lastly, Section 1.4 is one of the core part that we would like to achieve in this
thesis. We manage to classify the finite dimensional irreducible representations of
sln(C) solely by its highest weight vector (Theorem 1.4.1). In particular, we are capable
of classifying every irreducible representations of sl2(C) by a symmetric power of its
standard representation.
1.1 What is a Representation of sln(C)?
In this section, we define a representation of the semisimple Lie algebra sln(C). We
start off by defining sln(C) (Definition 1.1.3), as well as a representation of a Lie al-
gebra (Definition 1.1.8). To better understand the abstract idea of a representation of
sln(C), we provide some common examples which are often useful to be kept in mind.
Furthermore, the concept of ideal in a Lie algebra (Definition 1.1.13) is important to
show that sln(C) is a semisimple Lie algebra. Finally, we invoke a famous result regard-
ing a representation of semisimple Lie algbras – Preservation of Jordon Decomposition
Theorem where the argument for developing the adjoint representation of sln(C) is
based on.
1.1.1 Definitions
We intoduce all essential definitions to answer the title of the section. Understanding
the adjoint representation of sln(C) is critical for the rest of the Chapter 1.
Definition 1.1.1. A Lie algebra g is a vector space over a field F together with a
skew-symmetric bilinear map [ , ] : g × g → g, called the Lie bracket, satisfying the
Jacobi identity, that is, for all X,Y, Z in g, [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0.
Remark 1.1.2. We would like to remind the reader some properties of Lie bracket (also
known as the commutator of X and Y ), that are, for all a, b ∈ F , and all X,Y, Z ∈ g,
(L1) (Bilinearity) [aX + bY, Z] = a[X,Y ] + b[Y,Z], [Z, aX + bY ] = a[Z,X] + b[Z, Y ],
(L2) (Antisymmetry) [X,X] = 0,
6
(L3) The bracket operation satisfies the Jacobi identity.
In addition, if we apply L2 on X + Y , then we get the following:
(L2’) (Anticommutativity) [X,Y ] = −[Y,X].
Notice that L2 can be recovered from L2′ if the the characteristic of F is not 2 by taking
Y = X.
Definition 1.1.3. The set of n×n matrices with entries over a field F having trace 0
is called the special linear Lie algebra of order n which is denoted by sln(F ). Its set
notation isX =
a11 a12 . . . a1n
a21 a22 . . . a2n
......
. . ....
an1 an2 . . . ann
∈Mn(F )
∣∣∣∣∣∣∣∣∣∣Tr(X) =
n∑i=1
aii = a11 + a22 + . . .+ ann = 0
.
Lemma 1.1.4. Let X = (xij)1≤i,j≤n and Y = (yij)1≤i,j≤n be n× n matrices. Then,
Tr(XY ) = Tr(Y X).
Proof. Take any two n× n matrices X and Y , we get
Tr(XY ) =
n∑i=1
n∑k=1
xikyki =
n∑k=1
n∑i=1
ykixik = Tr(Y X).
QED
Now, we are going to introduce the core example of Lie algebra in this chapter.
Example 1.1.5. To start, we pick the basis for sl2(C). Consider the following basis
elements:
E =
[0 1
0 0
], F =
[0 0
1 0
], H =
[1 0
0 −1
]
with the relations
[H,E] = 2E, [H,F ] = −2F, [E,F ] = H.
One subtle thing to note is that this Lie algebra is a vector space of dimension three. It
is enough to check the Jacobi identity on the basis elements. Moreover, the set is closed
under the bracket operation since for all X,Y ∈ sl2(C), Tr(XY ) = Tr(Y X), so
Tr([XY ]) = Tr(XY − Y X) = Tr(XY )− Tr(Y X) = 0.
7
Example 1.1.6. Similarly, we can easily see that sln(C) is a Lie algebra of dimension
n2 − 1. First, let us pick a convenience basis to work with. Denote Ei,j as the n × nmatrix with 1 at the (i, j)-th entry and 0 at any other entries.
One can verify that, for i 6= j, the matrices
Ei,j
with 1 in the non-diagonal terms and the matrices
Hi,i+1 := Ei,i − Ei+1,i+1
with 1 and -1 in the consecutive diagonal terms form the basis of sln(F ) by the Spanning
Set Theorem in [Lay12]; there are n2− n+ (n− 1) = n2− 1 of them which are linearly
independent to one another. By using a similar argument in showing sl2(C) is closed,
we can see that sln(C) is closed under the bracket operation too.
Definition 1.1.7. Suppose g1 and g2 be Lie algebras over a field F . A Lie algebra
homomorphism is a linear map ϕ : g1 → g2 such that for all X,Y ∈ g1,
ϕ([X,Y ]) = [ϕ(X), ϕ(Y )].
Definition 1.1.8. A representation of a Lie algebra g on a vector space V is a Lie
algebra homomorphism ρ : g → End(V ) from a Lie algebra g to the Lie algebra of
endomorphism of V .
Example 1.1.9. Consider the standard representation of sl2(C) where V = C2 the
two-dimensional complex vector space. Then, we can take ρ to be the identity map
sending X ∈ sl2(C) to itself where the commutator relation is carried over trivially.
Example 1.1.10. Consider another representation of sl2(C) where V = Cn the n-
dimensional complex vector space. With Example 1.1.5 in mind, take the eigenvalue λ
of H with the biggest real part and pick a vector v in the eigenspace Vλ. With respect
to the basis v, Fv, . . . , Fn−1v of V , we can define ρ as follows:
ρ(H) =
n− 1 0 0 · · · 0
0 n− 3 0 · · · 0
0 0. . .
. . ....
......
. . . 3− n 0
0 0 . . . 0 1− n
ρ(E) =
0 n− 1 0 · · · 0 0
0 0 2(n− 2). . .
......
0 0 0. . . 0 0
0 0 0. . . (n− 2)2 0
......
. . . 0 n− 1
0 0 . . . 0 0 0
8
ρ(F ) =
0 0 0 · · · 0 0
1 0 0 · · · 0 0
0 1 0 · · · 0 0
0 0 1. . .
......
......
. . .. . . 0 0
0 0 . . . 0 1 0
.
This is done as Problem 1.55 in [EGHLSVY11].
Example 1.1.11. Consider the adjoint representation of sln(C), that is, we take
V = sln(C) to be itself. Then, we can take ρ to be the map sending X ∈ sln(C) to
ad(X) := [X, · ] ∈ End(sln(C)).
To check that ad is really a Lie algebra homomorphism, we need to check that for
all X,Y ∈ V ,
ad([X,Y ]) = [ad(X), ad(Y )].
Take an arbitrary Z ∈ sln(C),
[ad(X), ad(Y )](Z) = (ad(X)ad(Y )− ad(Y )ad(X))(Z)
= ad(X)ad(Y )(Z)− ad(Y )ad(X)(Z) by L1,
= [X, [Y,Z]]− [Y, [X,Z]] by definition,
= [X, [Y,Z]] + [Y, [Z,X]] by L1 and L2’,
= −[Z, [X,Y ]] by L3,
= [[X,Y ], Z]
= [ad([X,Y ]), Z],
as desired.
1.1.2 Semisimplicity of sln(C)
The definition of a semisimple Lie algebra involves the concept of an ideal of a Lie
algebra. We will soon see that sln(C) is not only semisimple, but is, in fact, simple.
Definition 1.1.12. A subspace h ⊂ g of a Lie algebra g is a subalgebra if it satisfies
the condition
[X,Y ] ∈ h for all X,Y ∈ h.
Definition 1.1.13. A Lie subalgebra h ⊂ g of a Lie algebra g is an ideal if it satisfies
the condition
[X,Y ] ∈ h for all X ∈ h, Y ∈ g.
Definition 1.1.14. The center Z(g) of a Lie algebra g is the subspace of g of elements
X ∈ g such that [X,Y ] = 0 for all Y ∈ g. Then, g is abelian if all brackets are zero,
or equivalently Z(g) = g.
9
Definition 1.1.15. A Lie algebra g is simple if dim g > 1 and it contains no
non-trivial ideals.
Definition 1.1.16. Let g denote a finite dimensional Lie algebra. The lower central
series of subalgebras Dng is defined inductively by
D1g = [g, g]
and
Dng = [g,Dn−1g]
giving the decreasing sequence
g ⊇ D1g ⊇ D2g ⊇ · · · .
In a similar manner, the derived series Dng is defined inductively by
D1g = [g, g]
and
Dng = [Dn−1g,Dn−1g].
giving the decreasing sequence
g ⊇ D1g ⊇ D2g ⊇ · · · .
Definition 1.1.17. A Lie algebra g is nilpotent if Dng = 0 for some n. A Lie algebra
g is solvable if Dng = 0 for some n.
Remark 1.1.18. Notice that Dng ⊂ Dng for every n. This implies that any nilpotent
Lie algebra is solvable too. When n = 1, Dg := D1g = D1g is called the commutative
subalgebra.
Definition 1.1.19. A Lie algebra g is semisimple if g has no non-zero solvable ideals.
Remark 1.1.20. Since a simple Lie algebra has no non-trivial ideal, it is automatically
semisimple. Furthermore, every semisimple Lie algebra is not solvable and hence not
nilpotent given Remark 1.1.18. Besides, any semisimple Lie algebra has a trivial center
because a non-trivial center is obviously a solvable ideal.
Proposition 1.1.21. The Lie algebra sl2(C) is semisimple.
Proof. The strategy is to prove that sl2(C) is simple. It suffices to show that each
basis elements generates the whole Lie algebra since we want to show that sl2(C)
has no non-trivial ideal. Then, the fact that sl2(C) is semisimple will follow from
Remark 1.1.20.
From the commutator relation in Example 1.1.5, we know immediately that E and
F are contained in the ideal generated by H by considering 12 [H,E] and 1
2 [H,F ]. It
is also not difficult to infer that H is contained in the ideal generated by E or F by
looking at [E,F ], so that the ideal generated by H is equal to the Lie algebra itself.
Finally, consider the ideal generated by an arbitrary element aE+bF +cH for some
a, b, c ∈ C by breaking into three cases:
10
(i) if a 6= 0, then
[[aE + bF + cH, F ], F ] = [[aE, F ] + [bF, F ] + [cH, F ]], F ]
= [aH − 2bF, F ]
= −2aF,
(ii) if b 6= 0, then
[[aE + bF + cH,E], E] = [−bH + 2cE,E] = −2bE,
(iii) if c 6= 0, then
[[aE + bF + cH,E], H] = [−bH + 2cX,H] = −4cE.
In either cases, by the preceding paragraph, they generate the entire Lie algebra.
QED
Proposition 1.1.22. The Lie algebra sln(C) is semisimple.
Proof sketch. A similar strategy as per the case of sln(C) is used here.
Take an nonzero element in a nontrivial ideal in sln(C). Following the notation in
Example 1.1.6, by suitable multiplication of Ei,j for i 6= j, we can see that the ideal
encompasses some basis elements Ei,j . Finally, we can see readily that the ideal is the
whole sln(C) by further multiplication. Hence, sln(C) is simple.
QED
Proposition 1.1.23. The adjoint representation of sln(C) as described in Example 1.1.11
is faithful.
Proof. Note that the kernel of the adjoint representation is the centre. But, we know
that sln(C) has trivial center from Remark 1.1.20 as we just shown that it is semisimple
in the previous proposition. Hence, the adjoint representation of sln(C) is faithful.
QED
1.1.3 General theory of semisimple Lie algebra
Furthermore, the semisimplicity of sln(C) allows us to decompose the representation of
sln(C) into direct sum of irreducible representation which we will do explicitly in the
later section. This is quoted as Theorem 9.12 in [FH91].
11
Complete Reducibility Theorem. Let V be a representation of the semisim-
ple Lie algebra g and W ⊂ V a subspace invariant under the action of g.
Then, there exists a subspace W ′ ⊂ V complementary to W and invariant
under g.
In addition, the representation of semisimple Lie algebra preserves the Jordan de-
composition. Let us recall the Jordan Decomposition Theorem.
Theorem 1.1.24 (Jordan Decomposition Theorem). Any endomorphism X of a com-
plex vector space V can be uniquely written in the form X = Xs + Xn where Xs is
diagonalizable and Xn is nilpotent, and the two commute.
Proof. Please refer to Theorem 4.3 in [BR02].
QED
Remark 1.1.25. However, the preservation of Jordan decomposition under a repre-
sentation does not hold in general. Consider a representation ρ : C→ End(C),
ρ : t 7→
[t t
0 o
].
The images ρ(t) are neither diaganoalizable and nilpotent. The situation is completely
different if the Lie algebra g is semisimple. This is quoted as Theorem 9.20 in [FH91].
Preservation of Jordan Decomposition Theorem. Let g be a semisim-
ple Lie Algebra. For any element X ∈ g, there exists Xs and Xn ∈ g such
that X = Xs +Xn and for any representation ρ : g→ End(V ), we have
ρ(X)s = ρ(Xs) and ρ(X)n = ρ(Xn).
1.2 The Adjoint Representation
In this section, we decompose the adjoint representation of sln(C) (Example 1.1.11)
into root spaces through Cartan decomposition of sln(C). Before doing this, we begin
by probing into the irreducible representation of sln(C) in Section 1.2.1. After that, we
study the finite dimensional irreducible representation of sl2(C) very closely in Section
1.2.2. Then, in Section 1.2.3, we generalize the idea of the highest weight vector in a
representation of sln(C) for larger n by suggesting an ordering on the set of roots.
12
1.2.1 Cartan decomposition of sln(C)
Let us return to the discussion of the finite dimensional representation of sln(C). From
Example 1.1.6, we found that Ei,i − Ei+1,i+1 spans a (n − 1)-dimensional subspace
h ⊂ sln(C) of all diagonal matrices. This subspace h is, in fact, a Cartan subalgebra
in the sense that it is a maximal abelian diagonalizable subalgebra of sln(C). Examine
how h acts on sln(C) through the adjoint representation.
From standard linear algebra, the commuting diagonalizable linear operators are
simultaneously diagonalizable. Moreover, any representation of sln(C) preserves the
Jordon Decomposition due to its semisimplicity as discussed in the previous section.
As a consequence, the property of h being diagonalizable is inherited by the action of
h on V , in other words, h acts diagonally on V !
By above analysis, we have the following definition.
Definition 1.2.1. Let g = sln(C). Suppose ρ : g → End(V ) is a representation of g
and h is a Cartan subalgebra of g. The weight for the action of h is a linear functional
α ∈ h∗ such that there exists some X ∈ V \ 0 satisfying for all H ∈ h,
ρ(H)(X) = α(H)X.
All X’s that satisfy the equation for a fixed α form a subspace called the weight space
associated to α, denoted by Vα. If h∗ consists of only constant maps, then the weight is
sometimes referred as an eigenvalue. Then, the corresponding weight space is just an
eigenspace. The dimension of Vα is the multiplicity of α in the representation.
Definition 1.2.2. Take V = g itself. We often refer a non-zero weight α of the adjoint
representation of g as a root of the Lie algebra g, that is, the linear functional α which
satisfies for all H ∈ h and all X ∈ gα,
ad(H)(X) := [H,X] = α(H)X.
Denote the set of roots by ∆. Subsequently, the corresponding weight spaces gα are called
the root spaces.
Remark 1.2.3. Note that ∆ ( h∗. There is a group acting on ∆. We will reveal the
group later.
In the light of this, we are able to decompose g = sln(C) into direct sum of eigenspace
associated to α. This root space decomposition called the Cartan decomposition,
sln(C) = h⊕(⊕α∈∆
gα
)where
gα = X ∈ g | for all H ∈ h, ad(H)(X) = α(H)X
13
which matches
sln(C) = h⊕(⊕
i 6=jCEi,j
).
Remark 1.2.4. Note that h is the space with zero weight although zero is not regarded
as a root conventionally. Also, h preserves all gα.
Example 1.2.5. For n = 3, it is easy to verify that
sl3(C) = H1,2 ⊕H2,3 ⊕ E1,2 ⊕ E1,3 ⊕ E2,1 ⊕ E2,3 ⊕ E3,1 ⊕ E3,2
= h⊕ gL1−L2 ⊕ gL1−L3 ⊕ gL2−L1 ⊕ gL2−L3 ⊕ gL3−L1 ⊕ gL3−L2 .
1.2.2 Representation of sl2(C)
In this part, we focus on the irreducible finite dimensional representation of sl2(C)
and use the highest weight vector to form the basis of the representation. Here, we
are able to deduce the direct relation between the weights and the dimension of the
representation.
A couple of definitions are required before we proceed.
Definition 1.2.6. Let ρ be a representation of a Lie algebra g acting on the space V.
A subspace W of V is called invariant if ρ(X)w ∈ W for all w ∈ W and all X ∈ g.
A subrepresentation of a representation V is a vector subspace W of V which is
invariant under g. An invariant subspace W is called proper if W 6= 0 and W 6= V.
A representation V is irreducible if there is no proper nonzero invariant subspace W
of V .
Example 1.2.7. From previous part, if we consider the adjoint representation of
sl2(C), then
sl2(C) = h⊕
g−2
⊕g2 = CH
⊕CF
⊕CE
which can be directly inferred from the commutator relation. This is one of its irre-
ducible finite dimensional representation .
Now, let V to be any finite dimensional irreducible representation of sl2(C). We are
now going to explore the representation theory of sl2(C) which lays down the foundation
of representation of sln(C). The span of H is a Cartan subalgebra in sl2(C). Since the
action of H on V is diagonalizable by Preservation of Jordan Decomposition Theorem
in Section 1.1.3, we can decompose
V =⊕
Vα
into eigenspaces Vα where α’s are the eigenvalues. But, we know more than that;
the eigenvalues are mod 2 congruent to one another and are integers. These are the
immediate consequences of the following theorems.
14
Theorem 1.2.8. The weights in a finite dimensional irreducible representation V of
sl2(C) form an unbroken string of numbers of the form α, α− 2, . . ., α− 2n for some
α ∈ C.
Proof. Suppose for some γ ∈ C, v ∈ Vγ , that is,
H(v) = βv.
We can see that E(v) ∈ Vγ+2 because
H(E(v)) = E(H(v)) + [H,E](v) = βE(v) + 2E(v) = (γ + 2)E(v).
By the same token, F (v) ∈ Vγ−2 as
H(F (v)) = F (H(v)) + [H,F ](v) = γF (v)− 2F (v) = (γ − 2)F (v).
Then, the subspace W =⊕
n∈Z Vγ+2n is invariant under the action of sl2(C). Since
V is irreducible, V = W . As V is finite dimensional, the sequence of weight must
terminate where the first term is chosen as α.
QED
Remark 1.2.9. In short, we can summarise the action of sl2(C) in the following dia-
gram with Vα the first element in the sequence:
Vα Vα−2 Vα−4 · · · .F
H
F
H
E
F
H
E E
The upshot of this is that the finite dimensionality of V guarantees the presence of
a maximal weight α in the sense that E(v) = 0 for all v ∈ Vα.
Theorem 1.2.10. An n-dimensional irreducible representation V for sl2(C) is spanned
by
v, F (v), F 2(v), ..., Fn−1(v)
where v lies in the highest weight space.
Proof. The existence of v is guaranteed by the finite-dimensionality of V . Start with
any eigenvector v′ of H associated to the eigenvalue α and apply E to v′.
Since V is finite dimensional, the process of applying E multiple times on v′ must
terminate after k+1 steps for some natural number k. Finally, we can take v = Ek(v′).
15
Using the irreducibility of V , it suffices to show that the subspace
W = Span(v, F (v), F 2(v), ...Fn−1(v)
)is invariant under sl2(C). Then, it follows the irreducibility of V and v 6= 0 that W = V.
By the nature of W , F preserves W since F sends Fm(v) to Fm+1(v) constituting those
spanning set elements. In addition, H also preserves W as the spanning set vectors are
also eigenvectors of H by the calculation in Theorem 1.2.8.
Next, we examine the action of E on sl2(C). First, we have E(v) = 0 which is
clearly in W . Then, E(F (v)) = F (E(v)) + [E,F ](v) = 0 + H(v) = nv where n is the
eigenvalues of v. Furthermore,
E(F 2(v)) = F (E(Fv)) + [E,F ](Fv) = nF (v) +H(F (v)) = (n+ (n− 2))F (v).
Inductively, we get
E(Fm(v)) = (n+ (n− 2) + . . .+ (n− 2(m− 1)))Fm−1(v)
= (mn−m(m− 1))Fm−1(v)
= m(n−m+ 1)Fm−1(v),
as desired.
QED
Remark 1.2.11. By the finite-dimensionality of V again, we have a lower bound on
the eigenvalue, in the sense that it has the smallest real part and there exist a smallest
m such that Fm(v) = 0. Subsequently, we obtain
0 = E(Fm(v)) = m(n−m+ 1)Fm−1(v).
Since Fm−1(v) 6= 0 by the choice of m, n must be m − 1 which is a non-negative
integer. By the virtue of Theorem 1.2.10, we can conclude that the eigenvalues are
integer-valued, differ by 2 from one another, and symmetric about the origin.
Corollary 1.2.12. The number of irreducible representations Ui in an arbitrary rep-
resentation V =⊕
i Ui of sl2(C) is exactly the sum of multiplicities of 0 and 1 as
eigenvalues of H, in other words, the sum of dimensions of the 0 and 1 eigenspaces in
the decomposition V =⊕
α Vα.
Generalization 1.2.13. The following are quoted as facts in [FH91] generalizing what
we have just shown in the case of sl2(C). We will verify them in Section 2.1.2 after we
introduce the Killing form of a Lie algebra.
Fact 1.2.14. For all sln(C),
(i) every root space gα will be one dimensional.
(ii) ∆ is symmetric about the origin, that is, if α ∈ ∆ is a root, then
−α ∈ ∆ is a root as well.
16
1.2.3 The highest weight vector in a representation
We will now formalise the definition of highest weight vector. Before that, we have
to define an ordering on the roots since the word “highest” suggests that the highest
weight vector varies with respect to the ordering we chose. From this vector, we can
obtain the other weight vectors in the irreducible representation.
Construction 1.2.15. For a general theory of sln(C), let explicitly
h =
a1 0 . . . 0
0 a2 . . . 0...
.... . .
...
0 0 . . . an
∈Mn(C)
∣∣∣∣∣∣∣∣∣∣a1 + a2 + . . .+ an = 0
be its Cartan subalgebra. We define a linear functional space h∗ on h by
h∗ =CL1, L2, . . . , LnL1 + L2 + . . .+ Ln
where
Li
a1 0 . . . 0
0 a2 . . . 0...
.... . .
...
0 0 . . . an
= ai, for i = 1, 2, . . . , n.
Next, we see that that gγ has an adjoint action on g. Take any E ∈ gα and F ∈ gγ,
we learn that ad(gγ) : gα → gα+γ because for any H ∈ h,
[H, [F,E]] = −[F, [E,H]]− [E, [H,F ]]
= −[F,−[H,E]]− [E, γ(H)F ]
= [F, [H,E]] + [γ(H)F,E]
= [F, α(H)E] + [γ(H)F,E]
= (α(H) + γ(H))[F,E].
Now, we choose an root ordering by picking a linear functional l such that
l
(n∑i=1
aiLi
)=
n∑i=1
ciai
withn∑i=1
ci = 0 and c1 > c2 > · · · > cn. Thus, this defines the positive root space for
∆+ to be gLi−Lj for i > j dividing the set of all roots ∆ = ∆+ ∪ ∆− into a positive
root set and a negative root set. We will adopt this convention throughout the thesis.
17
Remark 1.2.16. Notice that we are making a choice of root ordering here.
Example 1.2.17. Diagram 1.2.1 depicts the weight diagram for sl3(C). The thick ma-
genta line 1 divides ∆ into a positive root set ∆+ = L2 −L3, L1 −L3, L1 −L2 and a
negative root set ∆− = L2−L1, L3−L1, L3−L2. At the same time, the arrows show
the action 2 of ad(gL2−L1) on the root spaces.
0
L2 − L3
L2 − L1
L3 − L1
L3 − L2
L1 − L2
L1 − L3
0
0
0
L2
L1
L3
l = 0
Figure 1.2.1: The weight diagram for sl3(C) and the adjoint action of gL2−L1
on the root spaces.
Definition 1.2.18. A positive (respectively negative) root α ∈ ∆ is simple or primitive
if it cannot be expressed as a sum of two positive (respectively negative) roots.
Definition 1.2.19. Let V be any representation of g. A nonzero vector v ∈ V is called
a highest weight vector of V if that is both a weight vector for the action of h and
in the kernel of gα for all α ∈ ∆+.1As described before, this line depends on the linear functional you choose. For example, the thick
line is obtained by choosing a = 5, b = −1 and c = −4.2Note that there are only six one-dimensional root spaces for the adjoint representation of sl3(C).
18
Remark 1.2.20. This is exactly fulfilled by the highest weight vector of sl2(C).
Generalization 1.2.21. The next theorem is a generalization to the one in the case
sl2(C).
Theorem 1.2.22. The following statements are true for g = sln(C):
(i) Every finite-dimensional representation V of sln(C) possesses a highest weight
vector v.
(ii) The subspace W of V generated by the images of a highest weight vector v under
successive applications of root spaces gα for α ∈ ∆− is an irreducible representa-
tion.
(iii) An irreducible representation possesses a unique highest weight vector up to scalars.
Proof. The strategy is the same as before. First, for (i), the existence of v can be
examined by picking one vector from the Vα with α having “largest” l(α) in the sense
of Definition 1.2.19.
After that, for (ii), we just need to show that the subspace W of V spanned by
images of v under the subalgebra of sln(C) is preserved by all of sln(C) and hence must
be an irreducible representation. Let wn denote the any word of length n or less in the
elements of gγ for γ ∈ ∆− and Wn denote the vector space spanned by all wnv. We
argue by induction that EWn ⊂Wn for E ∈ gγ and γ ∈ ∆+. Write wn = Fwn−1 where
F ∈ gα and α ∈ ∆−. Then, EFwn−1 = FEwn−1 + [E,F ]wn−1 ⊂ Wn by inductive
hypothesis plus [E,F ] ∈ h. Hence, W , being the union of Wn , is a subrepresentation.
Suppose, for contrary, W is not irreducible which meansW = W ′⊕W ′′. By Fact 1.2.14
(i), the one dimensional weight space Wα must belong to W ′ or W ′′ and thus one of
the spaces is zero and the other equals to W .
Finally, for (iii), suppose v and w are the highest weight vectors but not scalar
multiples of each other in an irreducible representation Vα. Then, by part (ii), we
can form two irreducible subrepresentations generated by v and w under successive
applications of negative root spaces. Since Vα is an irreducible representation, the
subrepresentations must agree, and hence v and w must be scalar multiples of each
other.
QED
1.3 Full Picture of a Representation
In this section, we study the root lattice of a representation and weight lattice of the
Lie algebra. They are two fundamentally different lattices with one sitting in the other.
In section 1.3.1, we introduce a reflection, or rather an involution in the weight lattice.
The set of involutions form an interesting group called Weyl group. A point worth
19
noticing here is whatever we do depend on the choice of ordering. Section 1.3.2 is
devoted to deal with the multiplicity of weights of a representation. This is done by
treating a representation as a tableau.
1.3.1 Finding all weights
First, we want to know how the border vectors behave in the representation of g =
sln(C). Take an representative E in gα where α ∈ ∆− and apply it repeatedly to the
highest vector v associated to the eigenvalue α. However, the question is when the
string of eigenvalues will vanish.
Construction 1.3.1. By Fact 1.2.14 (2), g−α exists. The commutator h := [gα, g−α]
together with gα and g−α forms a subalgebra of g. Moreover, the commutator h has
dimension at most one since gα is one dimensional from Fact 1.2.14(1). As the adjoint
action of the commutator h sends each of gα and g−α into itself, we get the following
direct sum
sα := gα⊕
g−α⊕
[gα, g−α].
The claim is sα ∼= sl2(C). By picking a basis Eα ∈ gα and F−α ∈ g−α, it deter-
mines the unique Hα ∈ [gα, g−α] which has eigenvalue 2 on gα and −2 on g−α. In
other words, the two criteria – Hα ∈ [gα, g−α] and α(Hα) = 2 are solely character-
ize Hα. The adjoint representation of sln(C), therefore, inherits the integrality of the
eigenvalues under the action of Hα, that is, the eigenvalues are integral linear combi-
nations of Li.
Definition 1.3.2. The weight lattice of slnC is a Z–module of h∗ generated by
L1, . . . , Ln−1.
Definition 1.3.3. The root lattice of sln(C) is a Z–submodule of h∗ generated by the
roots of sln(C).
Form a weight lattice of sln(C), ΛW using the set of linear functionals l ∈ h∗.
Then, we see that ∆ also generate a root lattice Λ∆ ⊂ ΛW where explicitly,
ΛW = ZL1, · · · , Ln/(
n∑i=1
Li
)and
Λ∆ =
∑i
aiLi
∣∣∣∣∣ ai ∈ Z and∑i
ai = 0
/(n∑i=1
Li
).
Note that weight lattice and root lattice are not the same; weight lattice depends on the
Lie algebra whereas root lattice depends on the adjoint representation of the Lie algebra.
20
Returning to the preceding question, we introduce the following definition.
Definition 1.3.4. For a root α, an involution is defined as follows:
tα(γ) = γ − 2γ(Hα)
α(Hα)α = γ − γ(Hα)α
(as α(Hα) = 2), which is a reflection in the plane
Ωα = l ∈ h∗ : 〈Hα, l〉 = l(Hα) = 0.
Remark 1.3.5. Interestingly, these involutions tα generate a group which we call the
Weyl group W of the Lie algebra sln(C).
Definition 1.3.6. The closed Weyl chamber W associated to the ordering of the
roots is the real span of the roots α satisfying α(Hγ) ≥ 0 for every γ ∈ ∆+.
Remark 1.3.7. With respect to Construction 1.2.15,
W =∑
aiLi
∣∣∣ a1 ≥ a2 ≥ · · · ≥ an.
Geometrically, the vectors
L1, L1 + L2, L1 + L2 + L3, . . . , L1 + L2 + L3 + · · ·+ Ln−1
generate the edges of the cone W over an (n− 2)-simplex.
Notation 1.3.8. For an arbitrary (n − 1)-tuple of natural numbers (a1, · · · , an−1) ∈Nn−1, denote by
Γa1,··· ,an−1 = Γa1L1+a2(L1+L2)+···+an−1(L1+···+Ln−1),
an irreducible representation of sln(C) with the highest weight
a1L1 + a2(L1 + L2) + · · ·+ an−1(L1 + · · ·+ Ln−1).
Example 1.3.9. Figure 1.3.1 demonstrates how we utilize the involutions
tL1−L2 , tL1−L3 and tL2−L3
to find all the border vectors in the case of sl3(C). The yellow shaded part is W .
21
0
α
〈HL1−L2 , γ〉 = 0〈HL1−L3 , γ〉 = 0
〈HL2−L3 , γ〉 = 0
Figure 1.3.1: The Λ∆ of the representation Γα for sl3(C) in ΛW .
At the same time, we apply the same sl2(C) analysis to the border vectors to fill up the
inner diagram forming unbroken strings of eigenvalues.
Generalization 1.3.10. Generally, we can rewrite the eigenspace decomposition
V =⊕
Vλ
by grouping them in equivalence classes of eigenvalues
V =⊕
V[λ] =⊕[λ]
⊕n∈Z
Vλ+nα
as subrepresentations of V for sα. Choose λ and m ≥ 0 such that the set of weights
form unbroken string
λ, λ+ α, λ+ 2α, · · · , λ+mα
with the string of integers
λ(Hα), (λ+ α)(Hα), (λ+ 2β)(Hα), · · · , (λ+mα)(Hα)
22
which implies m = −λ(Hα) as (l+mα)(H) = λ(Hα)+2m and the string is symmetric
about zero. In other words, all the weights in Γγ are congruent to the highest weight
γ modulo the root lattice Λ∆ and lie completely in the convex hull of the images of γ
under the action of Weyl Group.
1.3.2 Multiplicity of the weights in an irreducible representation
This subsection will focus on the multiplicities of the weights in an irreducible represen-
tation and the dimension of an irreducible representation. To first find the multiplicity
of the weights, we can write the weights as a sum of the highest weight root and a few
primitive negative roots. Then, the multiplicity of the weight is exactly the number
of vectors that are independent to each other with different order of application of the
primitive negative roots to the highest weight root.
We will first introduce the definition of the tensor product of representations, n-th
symmetric power of a representation, and n-th exterior power of a representation.
Definition 1.3.11. Suppose ρV : g → End(V ) and ρW : g → End(W ) be two repre-
sentations of the Lie algebra g. The tensor product of two representations V and W
of a Lie algebra g is the space V ⊗W with
ρV⊗W (X) : g→ End(V ⊗W )
X 7→(v ⊗ w 7→ (ρV (X))(v)⊗ w + v ⊗ (ρW (X))(w)
),
or in other words, ρV⊗W (X) = ρV (X)⊗ Id+ Id⊗ ρW (X).
Remark 1.3.12. We can view a symmetric power of a vector space V,
SymnV := v ∈ V ⊗n | w(v) = v for all permutations w ∈ Sn
and an exterior power of a vector space V,∧nV := v ∈ V ⊗n | w(v) = −v for all permutations w ∈ Sn
as subsets of the tensor product of the vector space V. The n-th symmetric power of
a representations V of a Lie algebra g is the space SymnV → V ⊗n with an inherited
Lie algebra homomorphism
ρSymnV : g→ End(SymnV ).
The n-th exterior power of a representations V of a Lie algebra g is the space∧n V → V ⊗n with an inherited Lie algebra homomorphism
ρ∧n V : g→ End(∧n
V ),
One can check that the action of g on V⊗n preserves SymnV and
∧n V.
23
Example 1.3.13. To illustrate, we show that the weight L1 +L2 +L3 has multiplicity
two in the irreducible representation Γ1,1,0 of sl4 with the highest weight
1 · L1 + 1 · (L1 + L2) + 0 · (L1 + L2 + L3) = 2L1 + L2
contained in V⊗∧2 V . We notice that
L1 + L2 + L3 = 2L1 + L2 + (L2 − L1) + (L3 − L2).
Choosing e1⊗
(e1 ∧ e2) as the generator of the highest root space, we ought to
calculate
gL2−L1(gL3−L2(v)) = C · E2,1(E3,2(e1
⊗(e1 ∧ e2)))
= C · E2,1(E3,2(e1)⊗
(e1 ∧ e2) + e1
⊗E3,2(e1 ∧ e2))
= C · E2,1(0 + e1
⊗(E3,2(e1) ∧ e2) + e1
⊗e1 ∧ E3,2(e2))
= C · E2,1(0 + 0 + e1
⊗(e1 ∧ e3))
= C · E2,1(e1
⊗(e1 ∧ e3))
= C · (e2
⊗(e1 ∧ e3) + e1
⊗(e2 ∧ e3)), and
gL3−L2(gL2−L1(v)) = C · E3,2(E2,1(e1
⊗(e1 ∧ e2)))
= C · E3,2(e2
⊗(e1 ∧ e2) + e1
⊗(e2 ∧ e2))
= C · (e3
⊗(e1 ∧ e2) + e2
⊗(e1 ∧ e3)).
From the two equations, gL2−L1(gL3−L2(v)) and gL3−L2(gL2−L1(v)) are clearly linearly
independent to each other.
Otherwise, we can rely on the knowledge of Schur functor Sλ to determine the
multiplicity of weights. In order to achieve this, we need to introduce Young tableau.
Definition 1.3.14. A Young diagram, sometimes called a Young frame or Ferrers
diagram, associated to a partition λ = (λ1, . . . , λk) such that λ = λ1 + · · · + λk and
λ1 ≥ . . . ≥ λk is a collection of boxes, or cells, arranged in left-justified rows, with a
weakly decreasing number λi of boxes in the ith rows. The conjugate diagram with
conjugate partition λ is defined by flipping a Young diagram associated to λ over its
main diagonal from upper left to lower right, that is, interchanging rows and columns
in the Young diagram.
Definition 1.3.15. A Young tableau, or simply tableau, is a way of numbering each
box of a Young diagram that is weakly increasing across each row and strictly down
each column. A tableau is standard if the entries are the numbers from 1 to n, each
occurring exactly once.
24
Definition 1.3.16. Given a standard tableau with partition λ, we define two subgroups
of the symmetric group
P = g ∈ Sλ | g preserves each row and Q = g ∈ Sλ | g preserves each columnn.
After that, in term of group algebra CSλ, let
aλ =∑g∈P
and bλ =∑g∈Q
sgn(g) · eg
be two elements correlated to the subgroups. Till this end, we introduce a Young
symmetrizer cλ = aλ · bλ ∈ CSn.
Definition 1.3.17. For any finite dimensional complex vector space V , the Schur
functor or Weyl module, or simply Weyl’s construction SλV corresponding to λ is the
image, Im(cλ|V⊗n) of cλ restricted to V
⊗n.
The following theorem gives explicit construction of an irreducible representation.
Theorem 1.3.18. The representation Sλ(Cn) is the irreducible representation of sln(C)
with highest weight λ1L1 + λ2L2 + · · ·+ λnLn.
Proof. Please refer to the proof of Proposition 15.15 in [FH91].
QED
Remark 1.3.19. The proof of the theorem above in [FH91] tells us that the trace
of a matrix with respect to eigenspace decomposition corresponding to the eigenvalues
x1, . . . , xn on Sλ(Cn) is the Schur polynomial Sλ(x1, . . . , xn) =∑µKλµMµ where Kλµ
is the Kostka number giving the number of way filling the tableau of shape λ with µitimes of the natural number i and Mµ is the monomial xµ11 · · ·x
µii . From here, the
numbers Kλµ elucidate the multiplicities of possible weight spaces in the irreducible
representation with highest weight λ1L1 + λ2L2 + · · ·+ λnLn.
Remark 1.3.20. Moreover, we also know that the dimension of Sλ(Cn). Given an
irreducible Γa1,...,an−1 with highest weight
a1L1 + a2(L1 + L2) + . . .+ an−1(L1 + . . .+ Ln−1),
we can get our hands on computing
dim Γa1,...,an−1 = dim Sλ(Cn) =∏
1≤i<j≤n
(ai + · · ·+ aj−1) + j − ij − i
which can be found under Theorem 6.3(i) in [FH91].
25
1.4 Classifying the Finite Dimensional Irreducible Repre-
sentations
Eventually, we are able to classify the irreducible, finite dimensional representations of
sln(C) which the highest weight vector. For the case of sl2(C), we can say exactly what
are all the finite dimensional irreducible representations isomorphic to (Corollary 1.4.2).
Theorem 1.4.1. With respect to the ordering of the roots, for any α in the intersection
of the Weyl chamber W and the weight lattice ΛW , there exits a unique irreducible,
finite dimensional representation Γα of sln with highest weight α satisfying α(Hγ) ≥ 0
for each γ ∈ ∆+; this gives a bijection between W ∩ ΛW and the set of irreducible
representations of sln. Moreover, the weights of Γα will consist of those elements of the
weight lattice congruent to α modulo the root lattice Γ∆ and lying in the convex hull of
the set of points in h∗ conjugate to α under the Weyl group.
Proof. Note that the standard representation V ∼= Cn of sln(C) has highest weight
L1, the exterior power ∧kV is irreducible with highest weight L1 + . . . + Lk, and the
symmetric power SymkV is irreducible with highest weight akL1. The exterior power
∧kV and the symmetric power SymkV are irreducible because their weights occur with
multiplicity 1.
To prove the existence part, we can see that the irreducible representation Γa1,··· ,an−1
with highest weight (a1 + · · · + an−1)L1 + · · · + an−1Ln−1 shows up inside the tensor
product
Syma1V ⊗ Syma2(∧2V ) · · · ⊗ Syman−1(∧n−1V ).
The highest weight is obtained by taking the sum of highest weight of irreducible rep-
resentations appear in the tensor product.
Finally, if V and W are two finite dimensional irreducible representations of sln(C)
with highest weight vector v and w having the same weight α, then the vector (v, w) ∈V ⊕W is a highest weight vector associated to the weight α in that representation.
Suppose U ⊂ V ⊕W be subrepresentation generated by (v, w). Since U is irreducible
by construction, it follows that the two projection maps π1 : U → V and π2 : U → W
are isomorphisms, demonstrating the uniqueness part of the theorem.
QED
In particular, we can see a beautiful implication on classifying all irreducible repre-
sentation of sl2(C). This is a special case of Theorem 1.4.1.
Corollary 1.4.2. Any irreducible representation of sl2(C) is a symmetric power of the
standard representation V ∼= C2.
26
Proof. For the trivial one dimensional representation C, by considering the span of the
basis vector
(1
1
), we just get the representation V (0) with eigenvalue 0.
Next, if e and f are the standard basis for the standard representation C2, then
H(e) = e and H(f) = −f rendering us V = C2 = Cf⊕
Ce = V−1⊕V1 = V (1) a
representation with the highest eigenvalue 1. Similarly, we have a basis e2, ef, f2 for
the symmetric square Sym2C2. After a simple calculations such as
H(e · e) = H(e) · e+ e ·H(e) = 2e · e,H(e · f) = H(e) · f + e ·H(f) = 0,
H(f · f) = H(f) · f + f ·H(f) = 2f · f,
we found that V = Sym2C2 = Cf2⊕
Cef⊕Ce2 = V−2
⊕V0⊕V2 = V (2).
For the general case where V = Symn(C2), its basis are en, en−1f, . . . , fn and we
compute
H(en−if i) = H(en−i)f i + en−iH(f i)
= eH(en−i−1)f i + en−i−1H(e)f i + en−ifH(f i−1) + en−iH(f)f i−1
= e2H(en−i−2)f i + 2en−i−1H(e)f i + en−if2H(f i−2) + 2en−iH(f)f i−1
...
= (n− i)en−i−1H(e)f i + ien−iH(f)f i−1
= (n− i)en−if i − ien−if i
= (n− 2i)en−if i,
giving the eigenvalues n, n − 2, . . . ,−n. From Theorem 1.2.10, we know that an rep-
resentation having eigenvalues of H with multiplicity 1 is irreducible. Therefore, any
irreducible representation, V (n) with highest eigenvalues n is the nth symmetric power
of the standard representation V ∼= C2.
QED
27
28
Chapter 2
Weyl Group Acting on the Root
Lattice
In this chapter, we would like to pass the semisimple Lie algebra sln(C) to a root system
via a choice of diagonalizable Cartan subalgebra h. Previously, we learn that the root
space decomposition of the semisimple Lie algebra sln(C) is a result of the adjoint ac-
tion of the Cartan subalgebra h. Subsequently, we can define a so called “simple roots”
which are not sums of positive roots with an appropriate ordering of the underlying vec-
tor space of a root system (Definition 2.2.4). These roots serve a basis of the root lattice.
The purpose of this chapter is to develop the relation between the Weyl group and
the symmetric group. In contrast to Chapter 1, instead of viewing the Weyl group from
the point of view of adjoint representation of sln(C), we have the Weyl group generated
by reflections of the root system. Suppose ∆ is an abstract root system satisfying a
few axioms in a finite dimensional vector space E. In principle, we can form a subgroup
W = W(∆) of the orthogonal group on E generated by the reflections tα for α ∈ ∆.
Notice that the involution tα in Chapter 1 is a reflection. Here, W is the Weyl group of
∆. In our case, we specify the Weyl group W with the complex semisimple Lie algebra
sln(C), so it is sensible to express the Weyl group W as W(sln(C)). However, we will
mostly concern about the Weyl group W of sln(C) in this thesis.
In Section 2.1, we uncover a root system from the adjoint representation of sln(C).
In order to achieve this, we have to define a Killing form to get a notion of length
and angle in the representation, or more accurately, in the weight diagram formed by
the weights which are linear functionals. Despite an involution having already been
defined in Section 1.3, we have a second equivalent definition of an involution in terms
of the Killing form. The properties of root in adjoint representation of sln(C) (Theo-
rem 2.1.12) are the axioms of an abstract root system.
29
In Section 2.2, we see that the Weyl group of sln(C) as a particular instance of a
general Weyl group of an abstract root system. To speed up the calculation involving
the involutions, we provide a third equivalent definition of the involutions inW(sln(C)).
Theorem 2.2.10 is the most important theorem we proved in this chapter. In particular,
it provides an isomorphism between the Weyl group of sln(C) and the symmetric group
Sn where the symmetric group Sn is a group that we are extremely familiar with. We
can even write up the presentation of the the symmetric group Sn.
In Section 2.3, we provide S4∼=W(sl4(C)) as an illustration of a symmetric group
or a Weyl group acting on the root lattice of sl4(C). It serves as a perfect model to
understand the action. A few important observations can be made before we proceed
to Chapter 3 which we will see later that they occur in a more general setting.
2.1 Root System
In this section, we wish to identity a root system of the adjoint representation of sln(C).
Killing form is a key item we used to show the main properties take up by the roots.
Another main feature of a semisimple Lie algebra is the nondegeneracy of its Killing
form. This attribute plays an important role in gaining the properties we need for a
root system to hold in the adjoint representation of sln(C).
2.1.1 Killing form
We first define an inner product on the Lie algebra. Besides, we define the involution
in term of Killing form which is a sensible deed from the point of view of a abstract
root system.
Definition 2.1.1. Let g be a Lie algebra. The Killing form B of g is an inner product
defined by associating to each pair of elements X,Y ∈ g the trace of the composition of
their adjoint actions on g, that is,
B(X,Y ) = Tr(ad(X) ad(Y ) : g→ g).
Remark 2.1.2. It is obvious that the Killing form is a symmetric bilinear form. This
is because of the identity Tr(XY ) = Tr(Y X) for any endomorphisms X,Y of a vector
space and the bilinearity of the adjoint map.
Moreover, for all X,Y, Z ∈ g, the Killing form is associative with respect to bracket
operation, that is,
B([X,Y ], Z) = B(X, [Y, Z]).
However, for any endomorphisms X,Y , Z of a vector space,
Tr((XY − Y X)Z) = Tr(X(Y Z − ZY ))
30
since
Tr(Y XZ −XZY ) = Tr([Y ,XZ]) = 0.
Definition 2.1.3. Suppose V is finite dimensional vector space and B(·, ·) is an inner
product on V × V . The radical of B is defined by
rad B := v ∈ V | B(u, v) = 0 for all u ∈ V .
Then, B is nondegenerate if rad B = 0 .
Remark 2.1.4. Let V be a finite dimensional vector space. Define φ : V → V ∗
by 〈φ(v), u〉 = B(v, u) where 〈·, ·〉 is the pairing of the dual V ∗ with V . Notice that
ker φ = rad B. Thus, φ is an isomorphism if and only if B is nondegenerate.
Proposition 2.1.5. A Lie algebra g is semisimple if and only if its Killing form B is
nondegenerate.
Proof. Please refer to the proof of Proposition C.10 in [Ful97].
QED
Remark 2.1.6. Recall the adjoint representation of g = sln(C). Following from Re-
mark 2.1.4 and Proposition 2.1.5, we have an isomorphism φ : h → h∗ associating a
functional lH(X) = 〈H,X〉. To see this, suppose first Hα := [Eα, Fα] is the commutator
of an element Eα ∈ gα and Fα ∈ g−α. Then,
B(Hα, Hα) = B([Eα, Fα], Hα) by construction,
= B(Eα, [Fα, Hα]) by asscociativity of bracket,
= B(Eα, α(Hα)Fα) by definition,
= α(Hα)B(Eα, Fα) by bilinearity,
= 2B(Eα, Fα) by construction,
6= 0.
Otherwise, if B(Eα, Fα) = 0, then B(Hα, H) = 0 for every H ∈ h contradicting the
fact that B is nondegeneracy on h. Let the unique element Tα of h be the element such
that for all H ∈ h,
B(Tα, H) = α(H).
Hence, with a similar calculation as before, for all H ∈ h,
B(Tα, H) = α(H) = B(Hα, H)/B(Eα, Fα) = B(Hα/B(Eα, Fα), H)
which implies
Tα = Hα/B(Eα, Fα) = 2Hα/B(Hα, Hα).
31
We now define the Killing form B∗ on h∗ by B∗(γ, α) = B(Tγ , Tα).
Remark 2.1.7. For all root α,
B∗(α, α) = B(Tα, Tα) = B(2Hα
B(Hα, Hα),
2Hα
B(Hα, Hα)) =
2
B(Hα, Hα)
2
B(Hα, Hα)B(Hα, Hα)
=4
B(Hα, Hα).
Definition 2.1.8. For a root α, The involution tα(γ) can also be expressed in term
of roots by the formula
tα(γ) = γ − 2B∗(γ, α)
B∗(α, α)α.
Remark 2.1.9. This definition indeed agrees with Definition 1.3.4. To check, by Re-
mark 2.1.7,
2B∗(γ, α)
B∗(α, α)α =
2B(Tγ , Tα)
B(Tα, Tα)α = 2
B(Hα, Hα)
4B
(Tγ ,
2Hα
B(Hα, Hα)
)= B(Tγ , Hα) = γ(Hα),
as desired.
2.1.2 Properties of roots
Here, we are verifying the validity of Construction 1.3.1 and addressing Fact 1.2.14 at
the same time. The main theorem – Theorem 2.1.12 shows that the roots in the adjoint
representation of sln(C) constitute a root system.
The following proposition shows the orthogonality of root space and determine the
dimension of the subalgebra [gα, g−α] for α ∈ ∆.
Proposition 2.1.10. Suppose g = sln(C).
(i) The subspace gα and gγ are orthogonal if α+ γ 6= 0.
(ii) If α ∈ ∆, then [gα, g−α] is the one dimensional, with basis Tα.
32
Proof. (i) Note that for Xα ∈ gα, Yγ ∈ gγ and H ∈ h, by Remark 2.1.2,
0 = B([H,Xα], Y ) +B(Xα, [H,Yγ ]) = (α(H) + γ(H))B(Xa, Yγ).
If we choose α(H) + γ(H) 6= 0, then B(Xa, Yγ) = 0 for all Xa ∈ gα, and all
Yγ ∈ gγ , proving gα and gγ are orthogonal.
(ii) For Eα ∈ gα, F−α ∈ g−α and H ∈ h, we have
B([Eα, F−α], H) = B(Eα, [F−α, H]) = α(H)B(Eα, F−α) = B(Eα, F−α)B(Tα, H)
which gives
B([Eα, F−α]−B(Eα, F−α)Tα, H).
Thus, by degeneracy of B, [Eα, F−α] = B(Eα, F−α)Tα, as desired.
QED
We prove this theorem in response of Construction 1.3.1.
Theorem 2.1.11. Suppose g = sln(C).
(i) ∆ is symmetric about the origin, that is, if α ∈ ∆ is a root, then −α ∈ ∆ is a
root as well.
(ii) If α ∈ ∆ and Eα ∈ gα, then there exists F−α ∈ g−α such that Eα, F−α, and
Hα = [Eα, F−α] span a three dimensional subalgebra sα of g isomorphic to sl2(C).
via
Eα 7→
(0 1
0 0
), F−α 7→
(0 0
1 0
), Hα 7→
(1 0
0 −1
)
(iii) Every root space gα will be one dimensional.
Proof. (i) Suppose α ∈ ∆. If −α /∈ ∆, then, by Proposition 2.1.10(i), B(gα, gγ) = 0
for all γ ∈ h∗ contradicting the nondegeneracy of B.
33
(ii) By part (i) and Proposition 2.1.10(ii), we can find F−α ∈ g−α and
Hα = [Eα, F−α] 6= 0 with α(Hα) 6= 0. By suitably adjusting the scalars, they gen-
erate a subalgebra sα isomorphic to sl2(C).Using a key fact in Proposition 8.3(e) in
[Hum80], we know that we can choose F−α such that B(Eα, F−α) = 2/B(Tα, Tα).
Let Hα = 2Tα/B(Tα, Tα). We get
[Eα, F−α] = B(Eα, F−α)Tα =2
B(Tα, Tα)· B(Tα, Tα)Hα
2= Hα
by virtue of Proposition 2.1.10(ii). Furthermore, by the uniqueness of Tα and the
nature of Tα ∈ h,
[Hα, Eα] =
[2Tα
B(Tα, Tα), Eα
]=
2
α(Tα)[Tα, Eα] =
2α(Tα)
α(Tα)Eα = 2Eα,
as desired. By the same token, [Hα, F−α] = −2F−α. Moreover,
α(Hα) = B(Tα, Hα) = B
(Tα,
2TαB(Tα, Tα)
)= 2.
Hence, Eα, F−α, andHα span a three dimensional subalgebra isomorphic to sl2(C).
(iii) By part (i), we can pick Eα ∈ gα, F−α ∈ g−α, and Hα = [Eα, Fα]. Then, it is not
hard to realize that the subrepresentation
g′ = CFα ⊕ CHα ⊕⊕n>0
gnα
is invariant under the adjoint action of F−α, Hα, and Eα.
Next, we claim that ad(Hα) has trace 0 in its action on g′. To see this,
Tr(ad(Hα)) = Tr(ad([Eα, F−α])) = 0.
However, by the nature of Hα on g′, we have its trace
−2 + 0 +∞∑n=1
2n dim gnα = 0.
It follows that
dim gnα =
1, if n = 1,
0, if n = 2, 3, 4, . . . .
Thus, we obtain g = CEα which is one dimensional.
QED
34
This is the main theorem showing the four properties of the roots in adjoint repre-
sentation of sln(C) which then are used to generalise the notion of a root system.
Theorem 2.1.12. Suppose g = sln(C).
(i) The roots ∆ span h∗.
(ii) The roots of sln(C) are invariant under the Weyl group.
(iii) If α is a root, then the multiples of α which are roots are ±α.
(iv) For any α, γ ∈ ∆, 〈α, γ〉 := 2B∗(α,γ)B∗(γ,γ) ∈ Z.
Proof. (i) If ∆ fails to span h∗, then for each root α, there exists H ∈ h such that
α(H) = 0. Then, ad(H) = 0, that is H is in the center of g. But, g is semisimple.
Thus, H = 0.
(ii) Suppose γ and α are roots of g. It suffices to show that the roots congruent to λ
modulo α are invariant under the reflection tα. Consider the subrepresentation
U =⊕n∈Z
Vλ+nα
of the subalgebra sα. For some m 6= n, look at the string of roots
λ+ nα, λ+ (n+ 1)α, . . . , λ+mα
with the string evaluated at Hβ
λ(Hα) + 2n, λ(Hα) + 2n, . . . , λ(Hα) + 2m.
By the virtue of Theorem 2.1.11(i), λ(Hα) must be −(m + n) ∈ Z. Finally, for
k ≤ m− n,
tα(λ+ (n+ k)α) = λ+ (n+ k)α− (λ+ (n+ k)α)(Hα)α by definition,
= λ+ (n+ k)α− λ(Hα)α− (n+ k)α(Hα)α
= λ+ (n+ k)α+ (m+ n)β − (n+ k)2α by above,
= λ+ (m− k)α
which is still in U.
35
(iii) Suppose α is a root. We want to show that γ = nα is a root if and if n = ±1.
The “if” direction follows from 2.1.11(i). For the “only if” direction, assume that
γ = nα is a root. Consider applying γ to the unique elements Hγ and Hα; we get
two equations
γ(Hγ) = nα(Hγ) γ(Hα) = nα(Hα)
⇒ α(Hγ) =2
n⇒ γ(Hα) = 2n
which by part (2), 2n and 2n must be in integers. This restricted the possibility
of n to ±1, ±2 or ±12 . However, 2α and 4α are not in ∆ by Theorem 2.1.11(iii).
Thus, n can only be ±1.
(iv) This follows from part (iii).
QED
2.2 Reflections in the Root Lattice
In this section, we are trying to perceive the Weyl group purely from the point of view
of a reflection group acting on a subset of an euclidean space. Lastly, we find that
W(sln(C)) is isomorphic to Sn (Theorem 2.2.10).
2.2.1 Definition of a Weyl group
We introduce the axioms of an abstract root system .
Definition 2.2.1. An euclidean space E is a finite dimensional real vector space V
equipped with a positive definite symmetric bilinear form 〈α, γ〉.
Definition 2.2.2. A subset ∆ of the euclidean space E is called a root system in E
if the following axioms are satisfied:
(R1) ∆ is finite, spans E, and does not contain 0.
(R2) If α ∈ ∆, the only multiples of α in ∆ are ±α.
(R3) If α ∈ ∆, the reflection tα leaves ∆ invariant.
(R4) If α, γ ∈ ∆, the 〈γ, α〉 ∈ Z.
Remark 2.2.3. In the previous section, we verified that our root system of the adjoint
representation of sln(C) satisfies the axioms above.
Definition 2.2.4. A subset∑
of ∆ is called a base if
36
(B1)∑
is a basis of ∆,
(B2) Each root γ can be written as a sum of α ∈∑
with all non-negative or all
non-positive integral coefficients.
The roots in∑
are then called simple.
Remark 2.2.5. This definition agrees with Definition 1.2.18.
Theorem 2.2.6. Every abstract root systems ∆ has a base∑
.
Proof. Please refer to the proof of Theorem 10.1 in [Hum72]. QED
Now, let us formalise the definition of Weyl group. Recall that a reflection in a
euclidean space E is an invertible linear transformation sending all vectors orthogonal to
a hyperplane of codimension one into its negative and fixing the hyperplane pointwise.
Definition 2.2.7. Let ∆ be a root system in E. The Weyl group W of ∆ is generated
by the reflections tα for α ∈ ∆.
In Section 1.2.1, we obtain a root set from the adjoint representation of semisimple
Lie algebra sln(C). The reflections on the lattice formed by the roots generate the Weyl
group of sln(C). Denote the positive simple roots Li − Li+1 as αi. One can verify that
these roots, in fact, form a basis. To ease the manipulation involving reflections in the
Definition 1.3.4, we redefine the following.
Definition 2.2.8. To simplify our computation later, we define
〈αi, αj〉 :=
2, if i = j,
−1, if |i− j| = 1,
0, if |i− j| > 1.
and extend the definition bilinearly. With this convention, we redefine our reflecting
hyperplane as for α,
Ωα,= γ ∈ E | 〈γ, α〉 = 0
and the reflection
tα(γ) = γ − 〈γ, α〉α.
Remark 2.2.9. It is easy to show that these definitions agree those in Definition 1.3.4.
2.2.2 An isomorphism between W(sln(C)) and Sn
We will see that the Weyl group W of the complex semisimple Lie algebra sln(C) is,
in fact, the symmetric group Sn. The isomorphism between them helps us to perform
easier calculation of the action of W on the root lattice.
37
Theorem 2.2.10. The Weyl group, W(sln(C)) generated by reflections tαi associated
to a positive simple root αi is isomorphic to Sn.
Proof. Denote the transposition (i i+ 1) by si. First, note that Sn has a presentation
〈si, 1 ≤ i ≤ n− 1 | s2i = e, (sisi+1)3 = e, (sisj)
2 = e for |i− j| > 1〉.
We will show that the Weyl group satisfies the three relations in Sn. To achieve this,
we compute, for all γ ∈ ∆,
t2αi(γ) = tαitαi(γ)
= tαi(γ − 〈γ, αi〉αi)= γ − 〈γ, αi〉αi − 〈(γ − 〈γ, αi〉αi), αi〉αi= γ − 〈γ, αi〉αi − 〈γ, αi〉αi + 2〈γ, αi〉αi as 〈αi, αi〉 = 2,
= γ
= e(γ),
which implies t2αi= e where e is the identity element in Weyl group.
For each γ ∈ ∆ and |i− j| > 1,
tαitαj (γ) = tαi(γ − 〈γ, αj〉αj)= γ − 〈γ, αj〉αj − 〈(γ − 〈γ, αj〉αj), αi〉αi= γ − 〈γ, αj〉αj − 〈γ, αi〉αi as 〈αj , αi〉 = 0,
= γ − 〈γ, αj〉αj − 〈γ, αi〉αi + 〈〈γ, αi〉αi, αj〉αj as 〈αi, αj〉 = 0,
= γ − 〈γ, αi〉αi − 〈(γ − 〈γ, αi〉αi), αj〉αj= tαj (γ − 〈γ, αi〉αi)= tαj tαi(γ),
as desired.
Moreover, for each γ ∈ ∆ and |i− j| = 1,
tαitαj tαi(γ) = tαi(γ − 〈γ, αi〉αi − 〈γ, αj〉αj − 〈γ, αi〉αj)= γ − 〈γ, αi〉αi − 〈γ, αj〉αj − 〈γ, αi〉αj− 〈(γ − 〈γ, αi〉αi − 〈γ, αj〉αj − 〈γ, αi〉αj), αi〉αi
= γ − 〈γ, αi〉αi − 〈γ, αj〉αj − 〈γ, αi〉αj− 〈γ, αi〉αi − 〈γ, αj〉αi + 2〈γ, αi〉αi − 〈γ, αi〉αi
= γ − 〈γ, αi〉αi − 〈γ, αj〉αj − 〈γ, αi〉αj − 〈γ, αj〉αi
where the final expression is symmetrical in αi and αj . (One can also verify that
tαitαj tαi(γ) = tαj tαitαj (γ) by doing similar calculations on tαj tαitαj (γ)). Hence, this
38
implies that
tαitαj tαi = tαj tαitαj .
Giving these, we can easily construct a surjective group homomorphism, g from SntoW by sending si to tαi . Note thatW acts on ∆. In other word, the left multiplication
map by tαi induces a group homomorphism, f from W to the set Perm(∆) of bijective
map from ∆ to itself.
Now, the composition Sng−→W f−→ Perm(∆) defines a injective homomorphism. To
show this, we can take any si ∈ Sn since g is surjective and let it acts on ∆. It is easy to
see that the only w such that w(α) = α for all α ∈ ∆ is the identity element in Sn and
if we have the result of Theorem 3.2.1 which will be proved in the later chapter. Since
w(α) ∈ ∆+ for all α ∈ ∆+, there is no simple reflections such that `(wtαi) < `(w).
Then, w = e. Thus, we can infer that g is injective.
In the end, we conclude that g is an isomorphism, that is, W ∼= Sn.
QED
2.3 An example – S4
Let’s study how S4 acts on the root of the adjoint representation of sl4(C). By under-
standing this example, we will help to absorb the later material better.
First, we have the presentation of S4 :
〈si, 1 ≤ i ≤ 3 | s2i = e, (sisi+1)3 = e, (sisj)
2 = e for |i− j| > 1〉.
with all the elements
e, (12), (23), (34), (13), (14), (24), (12)(34), (13)(24), (14)(23),
(123), (234), (124), (134), (132), (143), (142), (243),
(1234), (1324), (1342), (1243), (1423), (1432).
Then, we pick out the generators such as s1 = (12) , s2 = (23), and s3 = (34). We
also denotes the the root by α1 = L1 − L2, α2 = L2 − L3, and α3 = L3 − L4. Now, an
action of S4 is set up such that an element w ∈ S4 acts on the root lattice by permuting
the subscript i of simple roots αi. It is not hard to establish that tα1 = s1, tα2 = s2,
and tα2 = s3.
Figure 2.3.1 show the root lattice of sl4(C). We have coloured lines on the root
lattice to indicate a hyperplane across the lattice. These hyperplanes are precisely
where the reflections happen. We can immediately see that there are six reflections in
W(sln(C)).
39
−α1 = L2 − L1
−α1 − α2 − α3 = L4 − L1
α2 = L2 − L3
−α3 = L4 − L3
−α1 − α2 = L3 − L1
−α2 = L3 − L2
α2 + α3 = L2 − L4
α1 + α2 + α3 = L1 − L4
−α2 − α3 = L4 − L2
α1 = L1 − L2
α1 + α2 = L1 − L3
α3 = L3 − L4
Ωα2+α3Ωα1+α2
Ωα2
Ωα1+α2+α3
Ωα1
Ωα3
Figure 2.3.1: The root lattice of sl4(C).
The table below demonstrates the action of a word w ∈ S4 on the roots. Notice
that we write every w in S4 in its minimal length. It is thrilling that we have this
table because a few interesting observations can be made here. They will be proved as
theorems in the next chapter.
We first observe that the word s3s2s1s2s3s2 is the longest element in the S4. It is
unique! Moreover, it is a product of six generators corresponding to the six roots in
the adjoint representation of sl4(C). The minimal length of a word w in S4 is exactly
the number of positive roots that are sent to negative roots by the action of the wordw
(Corollary 3.2.3).
Furthermore, we see that the reflections are elements in the form wsiw−1 for some
si and w in S4. The fact that they are reflections can be shown directly from the com-
putation in the table or see its action on Figure 2.3.1.
Before proving these observations occur in a more general setting, we need a suffi-
cient language to describe the elements in the Weyl group. This brings us to Chapter 3.
40
S4
(12)
(23)
(34)
(13)
(24)
(14)
(12)(
34)
(13)(
24)
Red
uce
dE
xpre
ssio
ns 1
s 2s 3
s 1s 2s 1
s 2s 3s 2
s 3s 2s 1s 2s 3
s 1s 3
s 2s 1s 3s 2
Refl
ecti
on
Yes
Yes
Yes
Yes
Yes
Yes
No
No
Act
ion
on
∆
α17→
−α1
α1
+α2
α1
−α2
α1
+α2
+α3
−(α
2+α3)
−α1
α3
α27→
α1
+α2
−α2
α2
+α3
−α1
−α2
α2
α1
+α2
+α3
−(α
1+α2
+α3)
α37→
α3
α2
+α3
−α3
α1
+α2
+α3
−α3
−(α
1+α2)
−α3
α1
α1
+α27→
α2
α1
α1
+α2
+α3
−(α
1+α2)
α1
+α2
−α3
α2
+α3
−(α
1+α2)
α2
+α37→
α1
+α2
+α3
α3
α2
α2
+α3
−(α
2+α3)
−α1
α1
+α2
−(α
2+α3)
α1
+α2
+α37→
α2
+α3
α1
+α2
+α3
α1
+α2
α3
α1
−(α
1+α2
+α3)
α2
−α2
S4
(14)(
23)
(1234)
(1324)
(1342)
(1243)
(1423)
(1432)
(123)
Red
uce
dE
xpre
ssio
ns 3s 2s 1s 2s 3s 2
s 1s 2s 3
s 1s 2s 3s 1s 2
s 2s 1s 3
s 1s 3s 2
s 2s 1s 3s 2s 1
s 3s 2s 1
s 1s 2
Refl
ecti
on
No
No
No
No
No
No
No
No
Act
ion
on
∆
α17→
−α3
α2
α3
−(α
1+α2)
α2
+α3
−α3
−(α
1+α2
+α3)
α2
α27→
−α2
α3
−(α
1+α2
+α3)
−(α
1+α2)
α1
−(α
1+α2)
α1
+α2
+α3
−(α
1+α2
+α3)
α37→
−α1
−(α
1+α2
+α3)
−α1
−(α
2+α3)
α1
+α2
α1
α2
α1
+α2
+α3
α1
+α27→
−(α
2+α3)
α2
+α3
−α2
α3
−α1
−(α
1+α2
+α3)
−(α
2+α3)
−α1
α2
+α37→
−(α
1+α2)
−(α
1+α2)
−(α
1+α2
+α3)
α1
−α3
−α2
α1
+α2
α3
α1
+α2
+α37→
−(α
1+α2
+α3)
−α1
−(α
1+α2)
−α2
α2
−(α
2+α3)
−α3
α2
+α3
S4
(234)
(124)
(134)
(132)
(143)
(142)
(243)
Red
uce
dE
xpre
ssio
ns 2s 3
s 1s 2s 3s 2
s 1s 2s 1s 3
s 2s 1
s 3s 1s 2s 1
s 2s 3s 2s 1
s 3s 2
Refl
ecti
on
No
No
No
No
No
No
No
No
Act
ion
on
∆
α17→
α1
+α2
α2
+α3
−α2
−(α
1+α2)
−(α
2+α3)
−(α
1+α2
+α3)
α1
+α2
+α3
α27→
α3
−α3
α2
+α3
α1
−α1
α1
+α2
−(α
2+α3)
α37→
−(α
2+α3)
−(α
1+α2)
−(α
1+α2
+α3)
α2
+α3
α1
+α2
−α2
α2
α1
+α27→
α1
+α2
+α3
α2
α3
−α2
−(α
1+α2
+α3)
−α3
α1
α2
+α37→
−α2
−(α
1+α2
+α3)
−α1
α1
+α2
+α3
α2
α1
−α3
α1
+α2
+α37→
α1
−α1
−(α
1+α2)
α3
−α3
−(α
2+α3)
α1
+α2
Tab
le2.
3.1:
Th
eac
tion
ofS
4∼ =W
(sl 4
(C))
onth
ero
ots
ofsl
4(C
).
41
42
Chapter 3
Combinatorics in the Weyl
Group of sln(C)
In this chapter, we will derive basic combinatorial properties in the Weyl group W to
a great extent. Begining with the work of W. Killing and E. Cartan, the Weyl group
of semisimple Lie algebras plays a crucial role in representation theory.
In 1934, H. S. M. Coxeter showed that every reflection group is a Coxeter group.
Here, a reflection group refers exactly to a group generated by reflections of a finite
dimensional Euclidean space. On the ground of W(sln(C) ∼= Sn, it is easy to realize
that the Weyl group is a perfect example of the Coxeter group. We are trying to un-
derstand the Coxeter groups algebraically, geometrically and combinatorially. Perhaps
the geometrical aspect will become more apparent in the next chapter when dealing
with the root set.
In Section 3.1, we define a Coxeter system which generalised the Weyl group. There-
fore, any properties held in a Coxeter system are going to be equally true in the Weyl
group. Lemma 3.1.12 represents a homomorphism in signed reflections in W(sln(C))
which can be relate to the action of the reflections on the root set ∆ of the sln(C) by
Theorem 3.1.13 . We also prove several useful properties taken by a minimal length
expressions.
In Section 3.2, we want to understand more about the elements in the Weyl group.
We have a stunning result which is the length of an element is the number of dis-
tinct positive roots sent to the negative root sets (Corollary 3.2.3). After that, we
show the existence of the longest word w0 in W(sln(C)). The most important attribute
it has is that we can obtain w0 from any w ∈ W(sln(C)) by a multiplication of an-
other w′ ∈ W(sln(C)), in other words, w0 = ww′ for any w ∈ W(sln(C)) and some
w′ ∈ W(sln(C)).
43
In Section 3.3, we define root sets for the expressions for elements in the Weyl group.
We then explore the root sets of reduced expressions.
3.1 Coxeter System
Coxeter system will supply the combinatorial structure of the Weyl group sufficiently.
We adopt a similar way to introduce a Coxeter system in [BB05].
3.1.1 Definitions
To begin with, we introduce essential definitions.
Definition 3.1.1. Given a group W and a set S, the pair (W,S) is called a Coxeter
system if the group W has a representationGenerators : S
Relations : (ss′)m(s,s′) = 1, with m(s, s′) ∈ N and m(s, s′) <∞,
where m(s, s) = 1, m(s, s′) = m(s′, s) ≥ 2 for s 6= s′ in S. If no such relations occurs for
a pair s, s′, the convention is m(s, s′) = ∞. In this case, the group W is the Coxeter
group and S is the generating set.
Remark 3.1.2. There is a universal property of Coxeter group W :
given a group G and f : S → G is a map such that
(f(s)f(s′))m(s,s′) = e for all (s, s′) with m(s, s′) 6=∞,
there exists a unique extension of f to a group homomorphism f : W → G.
Example 3.1.3. The symmetric group Sn is a typical example of Coxeter group.
Note that Sn has a presentation
〈si, 1 ≤ i ≤ n− 1 | s2i = e, (sisi+1)3 = e, (sisj)
2 = e for |i− j| > 1〉.
where si denotes the the transposition (i i+ 1).
Evidenced by Theorem 2.2.10 earlier, the Weyl group W of sln(C) is isomorphic to
Sn. Recall that tαi is a reflection associated to a simple positive root αi. When we view
W(sln(C)) as Sn, we regard the simple reflection tαi as si through the isomorphism in
Theorem 2.2.10. Therefore, any element tα for α ∈ ∆ can be expressed in terms of sifor notational brevity.
Throughout this chapter, it is convenient to keepW(sln(C)) or Sn in mind whenever
we talk about a general Coxeter system (W,S).
44
Notation 3.1.4. An element w ∈W has at least one expression w.
Remark 3.1.5. Given a Coxeter system (W,S). Denote S∗ as the free monoid gen-
erated by S, in other words, the set of concatenation of expressions in the alphabet S.
There is a surjective map ψ from S∗ to W. Therefore, an expression w is a lift of a
word w.
Example 3.1.6. In S4, w = s3s2s1s2 is an expression but s3s1s2s1, s1s3s2s1, and
s3s2s1s2 are representing the same element w. So, three of the them are expressions of
w with length of 4. We will make the definition of length precise in Definition 3.1.15.
Definition 3.1.7. Given a Coxeter system (W,S), define a reflection set T as follows;
T := wsw−1 | s ∈ S,w ∈W
containing elements called reflections.
Remark 3.1.8. The set T contains all elements that are conjugate to an element in
the generating set S of W . We can verify that t2 = e for t ∈ T . Easily, we can see that
S ⊆ T , so sometimes the elements of S are called simple reflections.
3.1.2 A bijection between reflections and roots
In this part, we try to link the reflections in W(sl(C)) with the roots of sln(C). More
precisely, Theorem 3.1.13 gives the required bijection.
Construction 3.1.9. Given an expression si1si2 . . . sin ∈ S∗, we continue to introduce
a couple of definitions: for 1 ≤ j ≤ n,
tij := (si1si2 . . . sij−1)sij (si1si2 . . . sij−1)−1 ∈ T,
the ordered n-tuple
T (si1si2 . . . sin) := (ti1 , ti2 , . . . , tin),
and
n(si1si2 . . . sin : t) := the number of times t appears in T (si1si2 . . . sin).
There are few interesting properties pertaining to ti such as
tijsi1si2 . . . sin = si1 . . . sij . . . sin
with sij omitted and
si1si2 . . . sij = tij tij−1 · · · ti1 .
Moreover, we want to study the group S(Ω) of all permutations of the set Ω =
T × −1, 1.
45
Definition 3.1.10. Define a bijective function πs from R to itself for each s ∈ S such
that
πs(t, ε) := (sts, εη(s : t))
where
η(s : t) :=
−1, if s = t,
+1, if s 6= t.
Proposition 3.1.11. If si1si2 . . . sin, with n minimal, then for all 1 ≤ j < k ≤ n,
tij 6= tik ∈ T (si1si2 . . . sin).
Proof. Suppose, for contrary, tij = tik for some j < k. But,
si1si2 . . . sin = tij tikw = tij tiksi1si2 . . . sin = si1 . . . sij . . . sik . . . sin
contradicting the minimality of si1si2 . . . sin .
QED
Lemma 3.1.12. (i) The mapping s 7→ πs extends uniquely to an injective homomor-
phism w 7→ πw from W to S(Ω).
(ii) For all t ∈ T , πt(t, ε) = (t,−ε).
Proof. (i) First, we want to utilise the universal property. Suppose we have (s, s) with
m(s, s′) 6=∞. Then,
π2s(t, ε) = πs(sts, εη(s : t)) = (sstss, εη(s : t)η(s : sts)) = (t, ε)
which implies that π2s = idR and πs ∈ S(Ω).
Next, suppose (s, s′) with m(s, s′) = p 6=∞. For the sake of notation simplicity, we
denote
si =
s′, if i is odd,
s, if i i is even.
We want to show that
(πsπs′)p = idR.
Now, we claim that
(πsπs′)p(t, ε) = (s2p . . . s1ts1 . . . s2p, ε
2p∏i=1
η(si : si−1 . . . s1ts1 . . . si−1)) = (t, ε).
46
The first component is clear since s2p . . . s1 = (ss′)p = e by assumption. Note that for
the second component,
ε
2p∏i=1
η(si : si−1 . . . s1ts1 . . . si−1) = ε(−1)n(s1s2...s2p:t).
To make sure that it is equal to ε, we need to show n(s1s2 . . . s2p : t) is even. Consider
T (s′ss′s . . . s′s︸ ︷︷ ︸2p
) and we found that
tp+i = ti
for 1 ≤ i ≤ p as (s, s′)p = e. Thus, if t appears in T (s′ss′s . . . s′s︸ ︷︷ ︸2p
), it must appear twice,
as desired.
By the universal property, we can extend uniquely the mapping s 7→ πs to a ho-
momorphism w 7→ πw from W to S(Ω). Let w = sinsin−1 . . . si1 be any arbitrary
expression. We see that
πw(t, ε) = πsinπsin−1. . . πsi1 (t, ε)
=
sin . . . si1tsi1 . . . sin , ε n∏j=1
η(sij : sij−1 . . . si1tsi1 . . . sij−1)
= (wtw−1, ε(−1)n(si1si2 ...sin : t)).
This is suggesting us to define
η(w−1 : t) := (−1)n(si1si2 ...sin :t)
which is well-defined since we just showed that n(s1s2 . . . sn : t) depends only on
w and t. Hence, we can write our new homomorphism as
πw(t, ε) = (wtw−1, εη(w−1 : t)).
This homomorphism is injective. Suppose w 6= e. Write w = sinsin−1 . . . si1 with
n minimal. Then, by Proposition 3.1.11, we know that
n(si1si2 . . . sin : ti) = 1
since T (si1si2 . . . sin) = (t1, t2, . . . , tn) and tk 6= tl for k 6= l.
It follows that the homomorphism is injective since
πw(tj , ε) = (wtjw−1,−ε) for 1 ≤ j ≤ n and so πw 6= idR.
47
(ii) We will prove by induction. Suppose t = si1 . Obviously,
πt(t, ε) = (ttt−1, εη(t : t)) = (t,−ε).
Now, let t = si1si2 . . . sin . . . si2si1 . Then, by inductive hypothesis together with part (i),
πt(t, ε)
= πsi1si2 ...sin ...si2si1 (si1si2 . . . sin . . . si2si1 , ε)
= πsi1πsi2 ...sin ...si2πsi1 (si1si2 . . . sin . . . si2si1 , ε)
= πsi1πsi2 ...sin ...si2 (si1si1si2 . . . sin . . . si2si1si1 , εη(si1 : si1si2 . . . sin . . . si2si1))
= πsi1πsi2 ...sin ...si2 (si2 . . . sin . . . si2 , εη(si1 : si1si2 . . . sin . . . si2si1))
= πsi1πsi2 ...sin ...si2 (si2 . . . sin . . . si2 , ε)
= πsi1 (si2 . . . sin . . . si2 ,−ε)
= (si1si2 . . . sin . . . si2si1 ,−εη(si1 : si2 . . . sin . . . si2))
= (t,−ε),
as desired.
QED
Theorem 3.1.13. Suppose Ω = T × +1,−1 and πw : Ω → Ω and let ∆ be the root
system of sln(C). The map φ : Ω→ ∆ defined by
(t,+1)→ γ ∈ ∆+ such that tγ = t,
(t,−1)→ γ ∈ ∆− such that tγ = t.
is bijective satisfying for all (t, ε) ∈ Ω and w ∈W,
φ(πw(t, ε)) = w(φ(t, ε)).
Proof. The map φ is surjective because every root is associated with a reflection. Fur-
thermore, the map φ is injective because the adjoint representation is faithful.
To check that φ(πw(t, ε)) = w(φ(t, ε)) for all (t, ε) ∈ Ω and w ∈ W, it suffices to
check it on the generator si, that is, for all si,
φ(πsi(t, ε)) = si(φ(t, ε)).
Note that
φ(πsi(t, ε)) = φ(sitsi, εη(si : t)) = si(γ) ∈ ∆εη(si:t)
and
si(φ(t, ε)) = si(γ)
48
for γ ∈ ∆ε. By definition, it is equivalent to see that
η(si : t) =
−1, if t = si,
1, if t 6= si,
and
si(γ) ∈
∆−ε, if γ = αi,
∆ε, if γ 6= αi.
But, γ = αi if and only if t = si. Thus, we obtain
φ(πsi(t, ε)) = si(φ(t, ε)).
for all si since the left hand side and the right hand side has the same image. QED
Remark 3.1.14. This theorem gives the precise bijection between signed reflections
and the set of roots, which intertiwes the action of W on both sets.
3.1.3 Minimal length expressions
We make a precise definition of length on the words inW(sln(C)). The strong exchange
condition tells us totally what the words can be under a certain condition (Corol-
lary 3.1.18) whereas the deletion condition tells us any expression can be reduced.
Definition 3.1.15. Given a Coxeter system (W,S), every element w ∈ W can be
expressed as a product of generators:
w = si1si2 . . . sin where sij ∈ S.
If n is the minimal among all such expressions for w, we write
`(w) = n
and n is the Bruhat length of w. Consequently, si1si2 . . . sin is a reduced expression
for w.
Remark 3.1.16. The length function on the word is always non-negative. If e is the
identity element in W , we may define `(e) := 0.
Theorem 3.1.17 (Strong Exchange Condition). Suppose w = si1si2 . . . sin with sij ∈ Sand t ∈ T . If `(tw) < `(w), then
tw = si1 . . . sij . . . sin
for some 1 ≤ j ≤ n.
49
Proof. Note that the condition
tw = si1 . . . sij . . . sin
is equivalent to show that
η(w, t) = −1.
If this is the case, then
t = si1si2 . . . sij . . . si2si1
from some j. It follows that tw = si1 . . . sij . . . sin .
First, suppose that `(tw) < `(w). We want to show that the formula from Lemma 3.1.12,
η(w, t) = (−1)n(si1si2 ...sin ,t) = −1.
In order to achieve a contradiction, assume η(w, t) = 1. Then,
π(tw)−1(t, ε) = πw−1πt(t, ε) by Lemma 3.1.12(1),
= πw−1(t,−ε) by Lemma 3.1.12(2),
= (w−1tw,−εη(w, t))
= (w−1tw,−ε) by assumption.
This implies that η(tw, t) = −1 which means n(tw, t) is odd. Consequently, we obtain
`(ttw) < `(tw)⇐⇒ `(w) < `(tw)
contradicting our assumption.
QED
Corollary 3.1.18. If w = si1si2 . . . sin is reduced and t ∈ T , then the following are
equivalent:
(1) `(tw) < `(w),
(2) tw = si1 . . . sij . . . sin , for some 1 ≤ j ≤ n,
(3) t = si1si2 . . . sij . . . si2si1 , for some 1 ≤ j ≤ n.
Furthermore, the index “ij” appearing in (ii) and (iii) is uniquely determined.
Proof. (1) ⇒ (2) It follows from the strong exchange property as proven before.
(2) ⇒ (1) This is obvious.
(2) ⇔ (3) It is also easy to see.
The uniqueness of the index ij follows immediately from Proposition 3.1.11.
QED
50
Inspired by these theorems, we establish the following definition.
Definition 3.1.19. We call the set of left associated reflections to w,
TL(w) := t ∈ T | `(tw) < `(w)
and the set of right associated reflections to w,
TR(w) := t ∈ T | `(wt) < `(w).
Remark 3.1.20. Note that TR(w) = TL(w−1).
Corollary 3.1.21. Let w = si1si2 . . . sin be reduced. Then,
TL(w) = si1si2 . . . sij . . . si2si1 | 1 ≤ j ≤ n
with |TL(w)| = `(w).
Proof. Write w = si1si2 . . . sin with `(w) = n. By Corollary 3.1.18,
TL(w) = si1si2 . . . sij . . . si2si1 | 1 ≤ j ≤ n.
Moreover, by Proposition 3.1.11, the elements in TL(w) are all different from each other.
QED
Corollary 3.1.22. For all s ∈ S and w ∈W , the following hold:
(1) s ∈ TL(w) if and only if some reduced expression for w begins with the letter s.
(2) s ∈ TR(w) if and only if some reduced expression for w ends with the letter s.
Proof. (1) (⇐=) If some reduced expression for w begins with the letter s, then s surely
satisfies the definition of TL(w).
(=⇒) Suppose `(sw) < `(w) with w = si1si2 . . . sin . Then, by Corol-
lary 3.1.18,
sw = si1 . . . sij . . . sin
for some ij . Since s is a Coxetor generator,
w = ssi1 . . . sij . . . sin ,
as desired.
(2) (⇐⇒) Apply the proof of (1) on the word w−1.
QED
Definition 3.1.23. Given a Coxeter syetem (W,S). For w ∈ W ,the left descent
set DL(w) is defined as the intersection TL(w) ∩ S. Similarly, the right descent set
DR(w) is defined as the intersection TR(w) ∩ S.
51
Proposition 3.1.24. (Deletion Condition) If w = si1si2 . . . sin and `(w) < k, then
w = si1 . . . sij . . . sik . . . sin
for some 1 ≤ j < k ≤ n.
Proof. Pick j maximal so that sijsij+1 . . . sin is not reduced. Obviously,
`(sijsij+1 . . . sin) < `(sij+1 . . . sin).
So, by Theorem 3.1.17,
sijsij+1 . . . sin = sij+1 . . . sik . . . sin
for some j < k ≤ n. Now, multiply si1si2 . . . sij−1 on both side, we get
si1si2 . . . sin = si1 . . . sij . . . sik . . . sin ,
as desired.
QED
Definition 3.1.25. A subexpression of an expression si1si2 . . . sin in a Coxeter group
W is an expression of the form sj1sj2 · · · sjk where i1 ≤ j1 < · · · < jk ≤ in.
Corollary 3.1.26. (i) Any expression si1si2 . . . sin contains a reduced expression for
w as a subexpression, obtainable by deleting an even number of alphabets.
(ii) Suppose w = si1si2 . . . sin = s′i1s′i2. . . s′in are two reduced expression. Then, the set
of letters appearing in the word si1si2 . . . sin equals the set of alphabets appearing
in s′i1s′i2. . . s′in.
(iii) S is a minimal generating set for W ; no Coxeter generator can be expressed in
terms of the others.
Proof. (i) This is an easy consequence from the Proposition 3.1.24.
(ii) Define S := si1 , . . . , sin. Choose j minimal such that sij is not in S. Then,
si1si2 . . . sij . . . si2si1 = s′i1s′i2 . . . s
′ij · · · s
′i2s′i1
for some j by Corollary 3.1.18. So,
sij = sij−1 . . . si1s′i1s′i2 . . . s
′ij . . . s
′i2s′i1si1 . . . sij−1
which reduced to an alphabets in S since every alphabets on the right is in S. However,
sij ∈ S contradicts the assumption that sij /∈ S.
(iii) This follows immediately from (ii) when w is a alphabet.
QED
52
Definition 3.1.27. Given a Coxeter system (W,S), denote by αs,s′ the alternating
expression ss′ss′s . . . of finite length m(s, s′). Then, the deletion of a factor of the
form ss is called a nil-move while the replacement of a factor αs,s′ by αs′,s is called a
braid-move.
Theorem 3.1.28. (Expression Property) Let (W,S) be a Coxeter group and w ∈W .
(i) Any expression si1si2 . . . sin for w can be transformed into a reduced expression
for w by a sequence of nil-moves and braid-moves.
(ii) Every two reduced expressions for w can be connected via a sequence of braid-moves.
Proof. Please refer to the proof of Theorem 3.3.1 in [BB05].
QED
Remark 3.1.29. This theorem will be our powerful tool to manipulate expressions in
our Coxeter system. In later chapter, it is extremely useful in proofs.
3.2 Expressions in Weyl Group of sln(C)
The aim of this section is to understand the expressions in W. The first part shows
that the longest expression in W is unique with special properties as mentioned in
Remark 3.2.12 and w0. The second part will deal with the finding the length of reduced
expression in W.
3.2.1 Reduced expression in W(sln(C))
Given an arbitrary expressions for elements in W, we can find its length by methods
introduced in this subsection.
Theorem 3.2.1. For all w ∈ W and α ∈ ∆+, `(wtα) < `(w) if and only if w(α) ∈ ∆−.
Proof. First, note that the condition `(wtα) < `(w) is equivalent to w = utα for some
expression u. Otherwise, `(wtα) won’t be less than `(w). In addition, if w = utαfor some expression u, then it is obviously `(wtα) < `(w). The proof will proceed by
induction argument on `(w) = k.
Base case: If k = 1, then w = si for some i. It follows that the statement is clearly
true.
Inductive hypothesis: When `(w) = n, `(wtα) < `(w)⇔ w(α) ∈ ∆−.
Inductive step: Assume `(w) = n+ 1.
(⇐=) Suppose w(α) ∈ ∆−. We want to show that `(wtα) < `(w). Write w = usijfor some sij ∈ S where `(u) = n. Therefore, w(α) = usij (α) ∈ ∆−. Now, sij (α) = −α
53
if and only if α = αij . If α = αij , then tα = sij . Hence,
`(wtα) = `(usijsij ) = `(u) < `(w).
Otherwise, w(α) = u(γ) ∈ ∆− for some γ ∈ ∆. By inductive hypothesis,
`(utγ) < `(u).
But, by Proposition 3.2.5, tγ = sij tαsij . Then,
`(u) > `(utγ) = `(usij tαsij ) = `(wtαsij )
which implies `(wtα) < `(u) + 1 = n+ 1 = `(w), as desired.
(=⇒) Suppose `(wtα) < `(w). We want to show that w(α) ∈ ∆−. Then,
w(α) = utα(α) = u(−α) = −u(α).
We claim that u(α) ∈ ∆+. This is because if u(α) ∈ ∆−, then by the inductive
hypothesis, u = vtα. However, this contradicts the minimality of w since w = vtαtα = v
and clearly `(v) < `(w).
QED
Remark 3.2.2. The essence of this theorem is that we can know that the length of the
word by the number of distinct positive roots sent to negative roots though its action.
This is proven in the following corollary.
Next, we associate a set of positive roots, α ∈ ∆+ | w(α) ∈ ∆− to each group
element w.
Corollary 3.2.3. For all w ∈W and α ∈ ∆+,∣∣α ∈ ∆+ | w(α) ∈ ∆−∣∣ =
∣∣tα ∈ T | `(wtα) < `(w)∣∣ = `(w).
Proof. The first equality follows from Theorem 3.2.1. In addition, the second equality
follows from Corollary 3.1.21 by looking at w−1. QED
Remark 3.2.4. Corollary 3.1.21 and Corollary 3.2.3 tell us that the sets
TR(w), tα ∈ T | `(wtα) < `(w), and α ∈ ∆+ | w(α) ∈ ∆−
has the same cardinality `(w).
Recall from Definition 2.2.8 that a reflection associated to a root α is defined as
tα(γ) := γ − 〈γ, α〉α .
Proposition 3.2.5. Suppose w ∈ W and γ ∈ ∆. If γ = w(α) for α ∈ ∆+, then
tγ = wtαw−1.
54
Proof. For any λ ∈ ∆,
wtαw−1(λ) = wtα(w−1(λ))
= w(w−1(λ)− 〈w−1(λ), α〉α)
= λ− 〈λ,w(α)〉w(α)
= λ− 〈λ, γ〉γ= tγ(λ).
QED
Theorem 3.2.6. There is a bijective correspondence between the set T of reflections
in W(sln(C)) and and the set ∆+ of simple positive roots.
Proof. We wish to define a bijective mapping ρ : ∆+ → T by sending γ = w(α) to
tγ = wtαw−1 for all w ∈ W and s ∈ S. First, note that the map ρ is well-defined. If
w(α) = u(α′), then for any λ ∈ ∆,
tw(α)(λ) = wsαw−1(λ)
= w(w−1(λ)− 〈w−1(λ), α〉α)
= λ− 〈λ,w(α)〉w(α)
= λ− 〈λ, u(α′)〉u(α′)
= u(u−1(λ)− 〈u−1(λ), β〉β)
= usα′u−1(λ)
= tu(α′)(λ).
Note that the only reflection that acts trivially on the roots is the identity. To see
this, first suppose t = si1 . . . sij−1sijsij−1 . . . si1 . Then, for all λ ∈ ∆,
λ = t(λ) = si1 . . . sij−1sijsij−1 . . . si1(λ)
⇐⇒sij−1 . . . si1(λ) = sijsij−1 . . . si1(λ)
⇐⇒γ = sij (γ) with γ = sij−1 . . . si1(λ).
But, sij (αij ) = −αij by definition. Therefore, sij must be the identity element e and
it follows that t = e. In particular, this tells us that tw(α) = tu(α′), as desired.
Moreover, it is surjective as every element in T can be written as
si1 . . . sij−1sijsij−1 . . . si1
with j minimal. If w(αij ) ∈ ∆−, then by Theorem 3.2.1, w has an expression ended
with sij which contradict the minimality of j. Therefore, w(αij ) must be in ∆+.
Besides, it is injective as if α and α′ are different positive roots, then tα 6= tα′ by
the definition of a root system (R2).
QED
55
3.2.2 The longest element in Weyl group of sln(C)
We mostly use results from [Hum94] to give us the desired property we want that the
longest element owns. This is explained in Remark 3.2.12.
Theorem 3.2.7. The Weyl group W(sln(C)) permutes the set of positive root sets
transitively.
Proof. Please refer to the proof of Theorem 1.4 in [Hum94]. QED
Remark 3.2.8. Note that positive root set is determined by our ordering on the roots.
If ∆+ is a positive root set, by a different ordering, ∆− could be equally a positive root
set as well.
Theorem 3.2.9. The following conditions on w ∈ W(sln(C)) are equivalent:
(1) w = e
(2) w∆+ = ∆+.
Proof. Please refer to the proof of Theorem 1.8 in [Hum94]. QED
Proposition 3.2.10. Let w0 be an element such that for each w ∈ W(sln(C)),
`(w0) = `(w) + `(w−1w0).
Suppose S is the generating set of W(sln(C)). Then, w0 satisfies the following condi-
tions:
(a) w0 is unique,
(b) w0 is an involution, and
(c) w0Sw0 = S.
Proof. (a) Suppose w0 and v0 are the elements such that for all w ∈W ,
`(w0) = `(w) + `(w−1w0) and `(v0) = `(w) + `(w−1v0).
Now, consider
`(w0) = `(v0) + `(v−10 w0) and `(v0) = `(w0) + `(w−1
0 v0).
The first equation tells us that `(w0) ≥ `(v0) whereas the second equation tells us that
`(v0) ≥ `(w0). For the two equations to hold true at the same time, we must have
`(v−10 w0) = 0 and `(w−1
0 v0)0, so that `(w0) = `(v0). But, this implies that w0 = v0
since v0 and w0 are inverses of each other.
56
(b) We want to show that w20 = e or, in other words, w0 is its inverse w−1
0 . This
follows immediately from the fact that w−10 is also a longest element in W and part (a).
(c) Suppose s ∈ S. Then, since s2 = e and w20 = e,
`(w0sw0) = `((sw0)−1w0)
= `(w0)− `(sw0) by assumption,
= `(w0)− `(w0) + `(s) by assumption,
= `(s)
= 1 by assumption,
as desired. This shows that w0sw0 is of length 1, in particular, w0sw0 ∈ S. Thus,
w0Sw0 ⊆ S. The reversed inclusion follows because w−10 = w0. Hence, w0Sw0 = S.
QED
Definition 3.2.11. The longest element in the Weyl group of sln(C) is the expression
of a word w0 satisfying
`(w0) = `(w) + `(w−1w0),
for each w ∈ W(sln(C)).
Remark 3.2.12. (The Existence of the Longest Element) Theorem 3.2.7 and Theo-
rem 3.2.9 together substantiates the simple transitivity of the permutation action of Won the set of positive root sets. The significance of the theorems is that there must exist
a unique word w0 with `(w0) = |∆+|.
By Theorem 3.2.1, w0 is distinguished by `(w0tα) < `(w0) for all α ∈ ∆. To find
the expression of w0, we can multiply a simple reflection si1 of length 1 on the left
successively until the process ceases. On the other hand, we can multiply any reduced
expression for w0 ∈ W by simple reflections on the right till we can’t do it anymore.
Finally, w0 is obtained.
From here, we can see that w0 = ww′ for some w′ ∈ W and w0 satisfies `(w0) =
`(w) + `(w′). In other words, for all w ∈ W, `(w0w) = `(w0)− `(w).
3.3 Root Sets for Expressions in the Weyl Group
This section colligates the reduced expressions in W with its root set which we define
below.
Definition 3.3.1. Let W denote the Weyl group of sln(C), ∆ denote the roots, ∆+ the
positive roots, and Σ the simple roots. For an expression si1 . . . sin, we define the left
root set of the expression
RootL(si1si2 . . . sin) = αi1 , si1 · αi2 , . . . , si1si2 . . . sin−1 · αin.
57
Here w · α is the root given by acting on α with the Weyl group element w. The right
root set RootR(si1si2 . . . sin) is defined analogously, reading from the right rather than
from the left.
We have the following basic facts.
Proposition 3.3.2. (i) The left and right root sets are related by
RootL(si1si2 . . . sin) = RootR(sinsin−1 . . . si1).
(ii) If si1si2 . . . sin is a reduced expression, then sinsin−1 . . . si1 is an another reduced
expression.
(iii) If si1si2 . . . sin is reduced, then the left root set RootL(si1si2 . . . sin) ⊂ ∆+.
(iv) The left root set RootL(si1 . . . sin) consists entirely of positive roots only if si1 . . . sinis a reduced expression.
(v) If two expressions si1si2 . . . sin = sj1sj2 . . . sjn are reduced, then the corresponding
left root sets are equal:
RootL(si1si2 . . . sin) = RootL(sj1sj2 . . . sjn).
(vi) For a reduced expression si1si2 . . . sin for w, we have
α ∈ RootL(si1si2 . . . sin) ⇐⇒ `(tαw) < `(w),
where `(w) denotes Bruhat length.
Proof. (i) This follows immediately from both definition of left and right root sets.
RootL(si1si2 . . . sik) = αi1 , si1 · αi2 , . . . , si1si2 . . . sin−1 · αin= RootR(sinsin−1 . . . si1).
(ii) It is immediate.
(iii) We have
RootL(si1si2 . . . sin) ⊂ ∆+
because si1si2 . . . sin is reduced. Otherwise, if there exists some 1 ≤ k ≤ n such that
si1si2 . . . sik−1(αik) ∈ ∆−,
58
then si1si2 . . . sik−1has an expression ending with sik contradicting the assumption that
si1si2 . . . sin is reduced.
(iv) We will prove by contrapositive. Suppose first w is not reduced, that is,
w has an expression si1 · · · sjsj · · · sim with at least one pair of sj adjacent to each
other for some j. This is because any expression for w can be reduced by a se-
quence of nil-moves and braid moves by Theorem 3.1.28. We might as well do all
possible braid moves first and collect the like term adjacent to each other. Con-
sider the left root set of w = si1 . . . siksjsjsik+3. . . sim . Note that si1 . . . sik · αj and
si1 . . . siksj · αj = −si1 . . . sik · αj are in the left root set of w. Since the two roots have
different sign, the left root set of w must contain a negative root.
(v) By Theorem 3.1.28, it suffices to look at two types of single braid moves one
by one. Suppose we have two expressions w1 = si1 . . . simskslsim+3 . . . sin , w2 =
si1 . . . simslsksim+3 . . . sin connected by the braid move sksl = slsk for |l − k| > 1.
Then, as sk · αl = αl and sl · αk = αk,
RootL(w1) = α1, si1 · αi2 , . . . , si1 . . . sim · αk, si1 . . . simsk · αl, . . . , si1 . . . sin−1 · αin= α1, si1 · αi2 , . . . , si1 . . . sim · αk, si1 . . . sim · αl, . . . , si1 . . . sin−1 · αin= α1, si1 · αi2 , . . . , si1 . . . sim · αl, si1 . . . sim · αk, . . . , si1 . . . sin−1 · αin= α1, si1 · αi2 , . . . , si1 . . . sim · αl, si1 . . . simsl · αk, . . . , si1 . . . sin−1 · αin= RootL(w2)
Second, suppose we have two words w1 = si1 . . . simskslsksim+3 . . . sin and
w2 = si1 . . . simslskslsim+3 . . . sin connected by the braid move skslsk = slskslfor |l− k| = 1. It suffices to see that since sk · αl = αk + αl = sl · ak, sksl · ak = αl and
slsk · al = αk,
si1si2 . . . sim · αk, si1si2 . . . simsk · αl, si1si2 . . . simsksl · αk= si1si2 . . . sim · αk, si1si2 . . . sim · (αk + αl), si1si2 . . . sim · αl= si1si2 . . . sim · αl, si1si2 . . . sim · (αk + αl), si1si2 . . . sim · αk= si1si2 . . . sim · αl, si1si2 . . . sim · sl · ak, si1si2 . . . simslsk · al
showing that they have the same root set.
(vi) (=⇒) Suppose α ∈ RootL(si1si2 . . . sin). Then, by definition,
α = si1si2 . . . sik−1· αik
for some 1 ≤ k ≤ n. By Proposition 3.2.5,
tαw = si1si2 . . . sik−1tαik
sik−1. . . si2si1w
= si1si2 . . . sik−1siksik−1
. . . si2si1si1si2 . . . sin
= si1si2 . . . sik−1siksik+1
. . . sin
59
where obviously `(tαw) < `(w).
(⇐=) Suppose `(tαw) < `(w). By Corollary 3.1.18, we see that
tα = si1si2 . . . sij . . . si2si1
for 1 ≤ j ≤ n. Hence, by Proposition 3.2.5 again,
tα = tsi1si2 ...sij−1·αij
which implies
α = si1si2 . . . sij−1 · αij ∈ RootL(si1si2 . . . sin),
as desired.
QED
60
Chapter 4
From the Weyl Group of sln(C) to
the Braid Group Bn
The aim of this chapter is to establish a correct notion of linear braid which will lead
to proving Conjecture 0.0.2, which is also known as Theorem 4.3.25. We would like to
remind the reader on the isomorphism between W(sln(C)) and Sn (Theorem 2.2.10).
From now onwards, we will frequently refer the Weyl group of sln(C) as W.
In Section 4.1, we introduce the braid group. We identify a few kinds of elements
such as the positive expressions, negative expression, and Garside generators. Note
that there is a canonical surjective homomorphism π from a braid group Bn to the
Weyl Group W by sending β ∈ σi, σ−1i to si. Subsequently, it makes sense to define
a positive lift and a negative lift of an expression for word in W.
In Section 4.2, we arrange the roots of an expression w for word inW into a positive
and a negative set with a certain rule on the expression for a braid which is a lift of w.
Then, we have some easy consequence from the definition.
In Section 4.3, we start by giving a proper definition of a linear braid. A few
important results are proven here. Corollary 4.3.14 states that all linear braids must
be minimal length lifts from the Weyl group whereas Corollary 4.3.15 states that all
reduced expressions for π(β) have lifts to expressions for β. Finally, the most prominent
theorem proven in this thesis is Theorem 4.3.25.
4.1 Definitions
Let us make some definitions to set up the chapter.
61
Definition 4.1.1. The braid group, Bn is a group with a presentation
〈σi, 1 ≤ i ≤ n− 1 | σiσi+1σi = σi+1σiσi+1, σiσj = σjσi for |i− j| > 1〉.
The generators σi are known as the Artin generators.
Remark 4.1.2. Notice that a difference between Bn and Sn is that the elements in Bnare not their own inverses.
Notation 4.1.3. An element β in the braid group Bn is called a braid. A braid β has
at least one expression β. Similarly to the symmetric group, we differentiate the braids
and their expressions with distinct symbols.
Example 4.1.4. In B3, β = σ1σ−12 σ1 has only one expression β = σ1σ
−12 σ1. But,
β = σ1σ2σ1 has two expressions, namely σ1σ2σ1 and σ2σ1σ2.
Definition 4.1.5. A positive expression is of the form
β+ = σi1σi2 . . . σim
for some i1, . . . , im and a negative expression is of the form
β− = σ−1i1σ−1i2. . . σ−1
im
for some i1, . . . , im.
Definition 4.1.6. Every element β ∈ Bn can be expressed as a product of generators:
β = σε1i1 σε2i2. . . σεnim .
If m is the minimal among all such expressions β for β, we write
`(β) = n
and n is the length (or sometimes `Artin) of β.
Definition 4.1.7. A preimage βw of w ∈ W under the surjective homomorphism π is
called a lift. A lift βw is positive if it has a positive expression. A lift βw is negative
if it has a negative expression.
Remark 4.1.8. Every w ∈ W has a positive lift and a negative lift. Any expression β
is a lift of w = π(β) in W.
Definition 4.1.9. A positive (respectively negative) lift is a positive (respectively nega-
tive) Garside generator if its positive (respectively negative) expression can be written
minimally such that `Artin(βw) = `(w). By convention, the identity element e in Bn is
considered as a positive and negative Garside generator.
Remark 4.1.10. Note that the Artin generators of Bn are also Garside generators.
62
Notation 4.1.11. Enlightened by Remark 4.1.8, given an element w ∈ W, we associate
the positive Garside generator β+w and the negative Garside generator β−w to it.
Example 4.1.12. Let s1s2 be the expression for w ∈ W. Then, β+w has a positive
expression σ1σ2 and β−w has a negative expression σ−11 σ−1
2 .
On the other hand, σ1σ1σ2 cannot be an expression for any positive Garside gener-
ator β+w since `(π(σ1σ1σ2)) = 1 6= 3 = `(β+
w ).
Remark 4.1.13. Note that a positive Garside generator β+w has a canonical mini-
mal length positive expression β+w although it is not unique. The canonical positive
expression can be written as a product of `(π(β)) generators. For example, σ1σ2σ1 and
σ2σ1σ2 are two different expressions representing the same positive Garside generator.
The same story applies to a negative Garside generator.
4.2 Positive and Negative Root Sets for Elements of the
Braid Group
Now, let Bn denote the braid group, with Artin generators σ±1i . Given an expression
β = σε1i1 σε2i2. . . σεkik ,
we define π(β) = si1 . . . sik to be the corresponding expression in W.
Definition 4.2.1. We define the left positive and negative root sets of the ex-
pression β by partitioning the left root set RootL(π(β)) according to the signs εi:
Root+L (β) = si1si2 . . . sil−1· αil | εl = 1,
Root−L (β) = si1si2 . . . sil−1· αil | εl = −1.
The right positive and negative root sets Root±R(β) are defined analogously, by
partitioning RootR(π(β)) according to the signs εi.
For an expression
β = σε1i1 σε2i2. . . σεnin ,
let
r(β) = σεnin σεn−1
in−1. . . σε1i1
denote the reverse of β, let
β−1 = σ−εninσ−εn−1
in−1. . . σ−ε1i1
,
and let
−β = σ−ε1i1σ−ε2i2
. . . σ−εnin.
Note that β−1 = r(−β) = −r(β).
63
Proposition 4.2.2. We have the following properties for root sets:
(i)
Root±L (β) = Root±R(r(β)).
(ii)
Root±L (β) = Root∓R(β−1).
(iii)
Root±L (β) = Root∓L (−β).
Proof. Write β = σε1i1 σε2i2. . . σεnik . These follow from the definitions easily. We have
Root+R(r(β)) = si1si2 . . . sil · αil−1| εl = 1 by definition,
= Root+L (β)
Root+R(β−1) = si1si2 . . . sil · αil−1| εl = −1 by the construction of β−1,
= Root−L (β)
Root+R(−β) = si1si2 . . . sil · αil−1| εl = −1 by the construction of −β,
= Root−L (β)
Their negative counterparts can be proved similarly.
QED
Proposition 4.2.3. Let β1, β2 be expressions in the braid group, and let β1 β2 denote
their concatenation. Then
Root±L (β1 β2) = Root±L (β1)⋃π(β1) ·Root±L (β2),
where π(β1)· denotes the action of the element of the Weyl group on roots.
Proof. Write β1 = σε1i1 σε2i2. . . σεmim and β2 = σδ1j1σ
δ2j2. . . σδnjn . Then, π(β1) = si1si2 . . . sim
and π(β2) = sj1sj2 . . . sjn . Compute
RootL(β1 β2) = αi1 , si2 · αi1 , . . . , si1si2 . . . sim−1 · αim ,si1 . . . sim · αj1 , si1si2 . . . simsj1 · αj2 , . . . , si1si2 . . . simsj1sj2 . . . sjn−1αjn
= αi1 , si2 · αi1 , . . . , si1si2 . . . sim−1 · αim ,π(β1) · αj1 , π(β
1) · (sj1sj2 · αj2), . . . , π(β
1) · (sj1sj2 . . . sjn−1 · αjn)
= RootL(β1)⋃π(β1) ·RootL(β2).
To show the lemma, we just need to sort the left roots into positive and negative
root set by imposing appropriate conditions on the power.
QED
64
4.3 Separated Root Sets and Linear Braids
Definition 4.3.1. We say that two subsets X,Y ⊂ ∆ of roots are separated if∑α∈X
nαα =∑γ∈Y
mγγ, with nα,mγ ≥ 0 for all α, γ =⇒ nα = mγ = 0 for all α, γ.
If X or Y is an empty set, then the sum will be zero by convention.
Remark 4.3.2. So in a pair of separated subsets, the non-negative cones defined by
the subsets intersect only at the origin.
We come to the fundamental definitions in this chapter.
Definition 4.3.3. An expression β = σε1i1 σε2i2. . . σεnin is left linear if Root+L (β) and
Root−L (β) are separated. Similarly, β is right linear if Root+R(β) and Root−R(β) are
separated.
Generally, the left root set and right root set of a braid are not equal. The aim of
following proposition is to find the relation between the left root set and right root set
of the same expression β.
Proposition 4.3.4. Let β be expression in the braid group. Then,
π(r(β)) ·RootL(β) = −RootR(β).
From the proposition, we can
Proof. Write β = σε1i1 σε2i2. . . σεnin . Take any α ∈ RootL(β). Then, α = si1si2 . . . sij−1 ·αij
for some 1 ≤ j ≤ n. Consider
π(r(β)) · α = sinsin−1 . . . si1si1si2 . . . sij−1 · αij= sinsin−1 . . . sij · αij= sinsin−1 . . . sij+1 · (−aij )= −sinsin−1 . . . sij+1 · aij ,
as desired. Hence, α ∈ −RootR(β) implies π(r(β)) ·RootL(β) ⊆ −RootR(β).
To show the other inclusion, it is equivalent to show −RootL(β) ⊇ π(β) ·RootR(β).
But, this can be done using a similar argument as before.
QED
Remark 4.3.5. A similar argument can show that −RootL(β) = π(β) ·RootR(β).
The following proposition allows us to move freely between the left root set and
right root set of an expression β.
65
Proposition 4.3.6. Suppose β is an expression in the braid group. If∑α∈Root+L (β)
nαα =∑
γ∈Root−L (β)
mγγ with nα,mγ ≥ 0 for all α, γ,
then there exists nα′ ,mγ′ for all α′, γ′ such that∑α′∈Root+R(β)
nα′α′ =
∑γ′∈Root−R(β)
mγ′γ′.
Proof. We just apply π(r(β)) to the equation∑
α∈Root+L (β) nαα =∑
γ∈Root−L (β)mγγ.
By the computation in Proposition 4.3.4, the equation become∑α∈Root+R(β)
−nαα′ =∑
γ′∈Root−R(β)
−mγγ′.
So, trivially, ∑α∈Root+R(β)
nαα′ =
∑γ′∈Root−R(β)
mγγ′
where we can simply take nα′ = nα and mγ′ = γ′.
QED
Remark 4.3.7. Proposition 4.3.6 is unequivocally true if we starts with zn equation
concerning only the right positive and negative root set, that is, if
∑α∈Root+R(β)
nαα =∑
γ∈Root−R(β)
mγγ with nα,mγ ≥ 0 for all α, γ,
then there exists nα′ ,mγ′ for all α′, γ′ such that
∑α′∈Root+L (β)
nα′α′ =
∑γ′∈Root−L (β)
mγ′γ′.
To see this, apply Remark 4.3.5.
Corollary 4.3.8. An expression β is left linear if and only if it is right linear.
Proof. This is an immediate consequence of Proposition 4.3.6 by considering the con-
trapositive of the statement above.
QED
Therefore, it makes sense to define the following.
Definition 4.3.9. We say a braid β is linear if β has a left (or, equivalently, right)
linear expression.
66
S3 B3 Root+L(β) Root−L (β) Garside generator Linear braid
s1 σ1 α1 ∅ Yes Yes
σ−11 ∅ α1 Yes Yes
s2 σ2 α2 ∅ Yes Yes
σ−12 ∅ α2 Yes Yes
s1s2 σ1σ2 α1, α1 + α2 ∅ Yes Yes
σ−11 σ−1
2 ∅ α1, α1 + α2 Yes Yes
σ1σ−12 α1 α1 + α2 No Yes
σ−11 σ2 α1 + α2 α1 No Yes
s2s1 σ2σ1 α2, α1 + α2 ∅ Yes Yes
σ−12 σ−1
1 ∅ α2, α1 + α2 Yes Yes
σ2σ−11 α2 α1 + α2 No Yes
σ−12 σ1 α1 + α2 α2 No Yes
s1s2s1 σ1σ2σ1 α1, α1 + α2, α2 ∅ Yes Yes
σ−11 σ2σ1 α1 + α2, α2 α1 No Yes
σ−11 σ−1
2 σ1 α2 α1, α1 + α2 No Yes
σ−11 σ−1
2 σ−11 ∅ α1, α1 + α2, α2 Yes Yes
= σ1σ−12 σ−1
1 α1 α1 + α2, α2 No Yes
σ1σ2σ−11 α1, α1 + α2 α2 No Yes
σ−11 σ2σ
−11 α1 + α2 α1, α2 No No
σ1σ−12 σ1 α1, α2 α1 + α2 No No
s2s1s2 σ2σ1σ2 α2, α1 + α2, α1 ∅ Yes Yes
σ−12 σ1σ2 α1 + α2, α1 α2 No Yes
σ−12 σ−1
1 σ2 α1 α2, α1 + α2 No Yes
σ−12 σ−1
1 σ−12 ∅ α2, α1 + α2, α1 Yes Yes
σ2σ−11 σ−1
2 α2 α1 + α2, α1 No Yes
σ2σ1σ−12 α2, α1 + α2 α1 No Yes
σ−12 σ1σ
−12 α1 + α2 α2, α1 No No
σ2σ−11 σ2 α2, α1 α1 + α2 No No
Table 4.3.1: The braids in B3.
Example 4.3.10. Table 4.2.1 tells us the linear braids in the braid group B3. It also
indicates which braids are the Garside generators. Note that we list all the expressions
in B3. So, some of the expressions represent the same element in B3. We can see that
linear braids with length 3 has two expressions. This is verified in a more general setting
later (Corollary 4.3.15).
Here are some basic facts about linear braids.
Definition 4.3.11. A subexpression of an expression β = σε1i1 σε2i2. . . σεnin is an expres-
sion of the form σεirirσεr+1
ir+1. . . σεsis , where 1 ≤ r ≤ s ≤ n.
Proposition 4.3.12. Any subexpression of a left linear expression β is left linear.
67
Proof. Let σε1i1 σε2i2. . . σεnin be a left linear expression β. Suppose, by contrary, there exists
some subexpression σεjijσεj+2
ij+2. . . σεkik for some 1 ≤ j, k ≤ n which is not left linear. Then,
by definition,
Root+L (σεjijσεj+2
ij+2. . . σεkik ) and Root−L (σ
εjijσεj+2
ij+2. . . σεkik )
are not separated. It follows that
π(σε1i1 σε2i2. . . σ
εj−1
ij−1)·Root+L (σ
εjijσεj+2
ij+2. . . σεkik ) and π(σε1i1 σ
ε2i2. . . σ
εj−1
ij−1)·Root−L (σ
εjijσεj+2
ij+2. . . σεkik )
are not separated. Note that
RootL(β) = RootL(σε1i1 σε2i2. . . σ
εj−1
ij−1)⋃π(σε1i1 σ
ε2i2. . . σ
εj−1
ij−1) ·RootL(σ
εjijσεj+2
ij+2. . . σεkik )⋃
π(σε1i1 σε2i2. . . σεkik ) ·RootL(σ
εk+1
ik+1σεk+2
ik+2. . . σεnin ).
Since the argument before suggests that Root+L (β) and Root−L (β) is not separated, β is
not a left linear expression.
QED
Theorem 4.3.13. Let β be a linear braid. Then, any left linear expression σε1i1 . . . σεmim
for β can be transformed into a left linear expression with minimal length by a sequence
of moves
σiσ−1i = σ−1
i σi = e,
σiσjσi = σjσiσjσiσjσ
−1i = σ−1
j σiσj , if |i− j| = 1
σiσ−1j σ−1
i = σ−1j σ−1
i σjσ−1i σ−1
j σ−1i = σ−1
j σ−1i σ−1
jσiσj = σjσiσiσ−1j = σ−1
j σi , if |i− j| > 1
σ−1i σ−1
j = σ−1j σ−1
i
Proof. Given any left linear expression β = σε1i1 . . . σεmim, look at the expression si1 . . . sim
for π(β) which is not necessarily reduced. By Theorem 3.1.28, there exists a sequence of
nil-moves and braid-moves to transform the expression for π(β) to a reduced expression
w.
We claim that every such moves lift to a valid move in Bn. Note that the nil-moves
sisi = 1 are lift to either σiσ−1i = 1 or σ−1
i σi = 1. If it was lifted to σiσi, then β will
not be linear in the first place by Proposition 4.3.12 as it contains a subexpression that
is not left linear.
Consider the following two braid moves, for |i− j| = 1, sisjsi = sjsisj is lifted to
σiσjσi = σjσiσj , σ−1i σ−1
j σ−1i = σ−1
j σ−1i σ−1
j
68
σiσjσ−1i = σ−1
j σiσj or σiσ−1j σ−1
i = σ−1j σ−1
i σj ,
and for |i− j| > 1, sisj = sisj is lifted to
σiσj = σjσi, σ−1i σ−1
j = σ−1j σ−1
i , σ−1i σj = σjσ
−1i or σiσ
−1j = σ−1
j σi.
If sisi+1si = si+1sisi+1 and sisj = sjsi for |i− j| > 1 are not lifted to these possible
linear cases, then β contains a subexpression which is not left linear which means β is
not left linear by Proposition 4.3.12 contradicting the assumption that β is left linear.
Since every nil-moves and braid-moves on expression for π(β) lift to the moves
stated in theorem, every left linear expression for β can be transformed into a left linear
expression with minimal length by a sequences of moves stated in the theorem. QED
As a corollary , we see that linear braids are all minimal length lifts for W.
Corollary 4.3.14. Suppose β = σε1i1 σε2i2. . . σεnin is a left linear expression for the linear
braid β. Then, `Artin(β) = `(π(β)). (In other words, linear braids must be minimal
length lifts from the Weyl group.)
Proof. First, we see that it is always the case that `Artin(β) ≥ `(π(β)) since Weyl group
has an extra relation, namely s2i = e.
We also have `Artin(β) ≤ `(π(β)) by Theorem 4.3.13 because any moves introduced
to reduce the expression for π(β) are lifted to a cancellation moves in the expression
for β.
Combining the two inequalities, we obtain `Artin(β) = `(π(β)), as desired.
QED
Corollary 4.3.15. Let β be a linear braid, and let π(β) ∈W be the image of β in the
Weyl group W. Then for all reduced expressions si1si2 . . . sin for π(β), there exist signs
η1, η2, . . . , ηn such that
β = ση1i1 ση2i2. . . σηnin .
(That is, all reduced expressions for π(β) have lifts to expressions for β.)
Proof. Let σε1j1σε2j2. . . σεrjr be a left linear expression for β ∈ Bn and let sp1sp2 . . . spn be
a reduced expression for π(β) ∈ W. We want to show that there exists η1, η2, . . . , ηn ∈±1 such that β = ση1i1 σ
η2i2. . . σηnin .
By Corollary 4.3.14, β = σδ1h1σδ2h2. . . σδnhn for some h1, h2, . . . , hn and δ1, δ2, . . . , δn. If
si1si2 . . . sin is another reduced expression for π(β), then si1si2 . . . sin and sp1sp2 . . . spnare connected by a sequence of braid-moves by Theorem 3.1.28. By Corollary 4.3.14
again, these moves are lifted to a valid braid-moves in the expression for the linear
braid β, so there exist signs η1, η2, . . . , ηn such that
β = ση1i1 ση2i2. . . σηnin .
for all reduced expressions si1si2 . . . sin for π(β). QED
69
Suppose β1 and β2 are linear braids. We ought to look for a condition on the
positive and negative root sets which tells us when the product β1β2 is linear.
Proposition 4.3.16. Suppose β1 and β2 are left linear expressions for linear braids.
Then, β1β2 is linear if and only if Root+L (β2)⋃Root−R(β1) and Root+R(β1)
⋃Root−L (β2)
are separated.
Proof. From Proposition 4.2.3, we know that β1 β2 is left linear if and only if
Root+L (β1)⋃π(β1) ·Root+L (β2) and Root−L (β1)
⋃π(β1) ·Root−L (β2)
are separated. By Proposition 4.3.4, we know that this is true if and only if
−Root+R(β1)⋃Root+L (β2) and −Root−R(β1)
⋃Root−L (β2)
are separated. But, it is equivalent to saying that
Root+L (β2)⋃Root−R(β1) and Root+R(β1)
⋃Root−L (β2)
are separated.
QED
If we consider the product of a positive Garside generators and a negative Garside
generators, we have the following special case of Proposition 4.3.16.
Corollary 4.3.17. Suppose β+x is a positive Garside generators and β−y is a negative
Garside generators. Then, β+x β−y and β−y β
+x are linear.
Proof. To show that they are linear, we need to prove that they have a left linear
expression. Consider, guaranteed by definition, the minimal length positive expression
for β+x and the minimal length negative expression for β−y . By Proposition 4.3.16, to
show that β+x β−y is linear, it suffices to check that
Root+L (β−y )⋃Root−R(β+
x ) and Root+R(β+x )⋃Root−L (β−y )
are separated. By construction of β+x and β−y , Root+L (β−y )
⋃Root−R(β+
x ) is empty.
In addition, Root+R(β+x )⋃Root−L (β−y ) consists of entirely positive roots by Proposi-
tion 3.3.2(iv). Hence, the linear condition is satisfied and β+x β−y is a left linear expres-
sion. Similar argument applies to β−y β+x .
QED
Proposition 4.3.18. Suppose w0 is the longest element inW. If w0 = xy with `(w0) =
`(x) + `(y), then β+w0
= β+x β
+y and β−w0
= β−x β−y .
Proof. This is clear from the definition of Garside generators. QED
70
Proposition 4.3.19. The following are equivalent:
(1) β has an expression β+x β−y for some x and y in W.
(2) β has an expression β−u β+v for some u and v in W.
Proof. (1) =⇒ (2) Let w0 be the element of longest length in the Weyl group W and
β+w0
be the associated positive Garside generator. Subsequently, by the property of the
element of longest length, we can find z ∈ W such that z = w0x−1 with its positive
Garside generator β+z and v ∈ W such that v = w0y with its positive Garside generator
β+v . Then, by Proposition 4.3.18,
β = β+x β−y
= (β+z )−1β+
z β+x β−y
= (β+z )−1β+
w0β−y
= (β+z )−1β+
v ,
where we have written β in the required arrangement. This proves that (1) =⇒ (2).
To get (1) ⇐= (2), we apply a similar argument. QED
Definition 4.3.20. We define a set E+L (β) for a braid as following:
E+L (β) := σi | σiβ is not linear.,
and similarly, we can define a set E+L (β) for a braid as following:
E−R (β) := σ−1i | βσ
−1i is not linear.,
Recall the definition of the descents of Weyl group W. Let S be the generating set
for W. For w ∈ W, the left descent set DL(w) := TL(w) ∩ S and the right descent set
DR(w) := TR(w) ∩ S.
Definition 4.3.21. Analogously, a left descent set DL(β) for a braid is defined as
DL(β) := σεj | β has a minimal length expression β = σεjσε1i1σε2i2 . . . σ
εmim.
and a right descent set DR(β) for a braid is defined as
DR(β) := σεk | β has a minimal length expression β = σε1i1 σε2i2. . . σεmim σ
εkk . .
Lemma 4.3.22. Let β be a left linear expression for β and β+x β−y be its subexpression
with DR(β+x ) ∩ DL(β−y ) = ∅ such that there is no internal cancellation of terms in
the subexpression. Suppose β+x β−y = β−u β
+v with DR(β−u ) ∩ DL(β+
v ) = ∅. Then, the
expression β′ given by the replacement of β+x β−y by β−u β+
v in the expression β for β is
left linear.
71
Proof. We prove by induction of the length of β+x and β−y . It actually suffices to check
the statement when β+x = σi and β−y = σ−1
j . If σiσ−1j = σ−1
j σi, the claim is obvious.
Otherwise, σiσ−1j = σ−1
j σ−1i σjσi. So that, β = wσiσ
−1j z and β′ = wσ−1
j σ−1i σjσiz.
Then,
Root+L (β) = Root+L (w)⋃π(w) · αi
⋃π(wσiσ
−1j ) ·Root+L (z),
Root−L (β) = Root+L (w)⋃π(w) · αi + αj
⋃π(wσiσ
−1j ) ·Root+L (z),
Root+L (β′) = Root+L (w)⋃π(w) · αi,−αj
⋃π(wσiσ
−1j ) ·Root+L (z),
and
Root−L (β′) = Root+L (w)⋃π(w) · αj , αi + αj
⋃π(wσiσ
−1j ) ·Root+L (z).
Note that these sets differ only in that β′ has two more roots than β in its root set.
Suppose that Root+L (β′) and Root−L (β′) are not separated. Acting by (π(w))−1, we
get a equation relating Root+L (β′) and Root−L (β′) of the form
r+ + n1αi + n2(−αj) = r− +m1αj +m2(αi + αj)
for n1, n2,m1,m2 ∈ N, where r+ is some linear combination of other roots in Root+L (β′)
and r− is some linear combination of other roots in Root−L (β′). After rearranging, we
get
r+ + (n1 +m1 + n2)(αi) = r− + (m1 + n2 +m2)(αi + αj)
for n1, n2,m1,m2 ∈ N where r+ is some linear combination of other roots in Root+L (β′)
and r− is some linear combination of other roots in Root−L (β′). After applying π(w)
to the equation above, it yields a non-trivial relation between Root+L (β) and Root−L (β)
which means β is not linear. A contradiction.
QED
Proposition 4.3.23. Given a braid β with an expression β = β+x β−y with DR(β+
x ) ∩DL(β−y ) = ∅ such that there is no internal cancellation of terms in the expression where
β+x = σj1σj2 . . . σjt is a positive Garside generator from W and β−y = σ−1
l1σ−1l2. . . σ−1
lkis
a negative Garside generator from W.
Then, every elements in E+L (β) under the map π are exactly the elements in DL(π(β+
x )),
in other words, π(E+L (β)) = DL(π(β+
x )).
Proof. To show that DL(π(β+x )) ⊂ π(E+
L (β)), suppose si ∈ DL(π(β+x )), that is, π(β+
x )
has an expression starting with si. We want to show that si ∈ π(E+L (β)), that is σi or
σ−1i is in E+
L (β). Since π(β+x ) has an expression starting with si, then β+
x also has a
minimal length positive expression starting with σi, that is β+x = σiβ
′+ for some σiβ′+
with `(β′) = `(β+x )− 1. Then, β has an expression
σiβ+x β−y = σiσiβ
′+β−y
72
which is not left linear as αi − αi = 0. But, by Lemma 4.3.22, β cannot have a left
linear expression. So, the claim is proved.
For the converse, suppose σiβ+x β−y is not left linear. Then, it follows that σiβ
+x is
not left linear by Corollary 4.3.17. Subsequently, σiβ+x is not a positive lift from W.
Then,
`(π(σiβ+x )) < `(π(β+
x )).
As π(σiβ+x ) = siπ(β+
x ), we obtain si ∈ DL(π(β+x )) by Corollary 3.1.22.
QED
Remark 4.3.24. Given the same condition in Proposition 4.3.23, similarly one can
also show that π(E−R (β)) = DR(π(β−y )).
The following is one of the main theorems in this document. It relates the linear
braids to the Garside generators.
Theorem 4.3.25. The following are equivalent:
(1) β is linear.
(2) β has an expression β+x β−y for some x and y in W.
(3) β has an expression β−u β+v for some u and v in W.
Proof. (2) ⇐⇒ (3) It follows from Proposition 4.3.19.
(1) =⇒ (2) We will prove by induction on the length of β. Suppose β is linear.
Base case: If β = σεi is of length 1, then it is in the desired form.
Inductive hypothesis: Suppose that when `(β) ≤ n we can write β = β+x β−y , where
β+x = σj1σj2 . . . σjt is a positive Garside generator from W and β−y = σ−1
l1σ−1l2. . . σ−1
lk
is a negative Garside generator from W when β = σε1i1 σε2i2. . . σεnin is a minimal length
expression for `(β).
Inductive step: Suppose now β is a linear braid with `(β) = n+ 1 having a minimal
length expression β = σε1i1 σε2i2. . . σ
εn+1
in+1.
Assume first ε1 = 1. Then, β = σi1β′ with σε2i2 . . . σ
εn+1
in+1being an expression β′ and
`(β′) < `(β). Notice that any subexpression of β is left linear by Proposition 4.3.12. β′
is also a minimal length expression since otherwise β would not be a minimal length
expression at first. So, by inductive hypothesis, β+x β−y is an expression for β′. Note
that σi1 /∈ E+L (β′). If not, β is not linear. By Proposition 4.3.23, σi1 /∈ DL(β+
x ). Hence,
σi1β+x is a positive Garside generator and β has an expression σi1β
+x β−y which is a
product of a positive Garside generator and a negative Garside generator.
On the other hand, if ε = −1, we can use a similar induction argument on (1) =⇒ (3).
So that, we can write β = β−u β+v where β−u is a negative Garside generator and β+
v is a
73
positive Garside generator. By the earlier proof of (2)⇐⇒ (3), we can get (2) eventually.
(2) =⇒ (1) This follows from Corollary 4.3.17.
QED
74
Chapter 5
Presentations for the Braid
Group and their Length
Functions
Naturally, we have three families of generators to consider as the generators of the braid
group. Recall that the definition of the braid group Bn:
Bn := 〈σi, 1 ≤ i ≤ n− 1 | σiσi+1σi = σi+1σiσi+1, σiσj = σjσi for |i− j| > 1〉.
Those σi’s are known as the Artin generators. The other two generator candidates,
namely the Garside generator and the linear braids are met in Chapter 4. Say in B3,
we are claiming that the sets
σ1, σ−11 , σ2, σ
−12 , σ1σ2, σ
−11 σ−1
2 , σ2σ1, σ−12 σ−1
1 , σ1σ2σ1, σ−11 σ−1
2 σ−11 , σ2σ1σ2, σ
−12 σ−1
1 σ−12
and
σ1, σ−11 , σ2, σ
−12 , σ1σ2, σ
−11 σ−1
2 , σ1σ−12 , σ−1
1 σ2, σ2σ1, σ−12 σ−1
1 , σ2σ−11 , σ−1
2 σ1, σ1σ2σ1,
σ−11 σ2σ1, σ
−11 σ−1
2 σ1, σ−11 σ−1
2 σ−11 , σ2σ1σ2, σ
−12 σ1σ2, σ
−12 σ−1
1 σ2, σ−12 σ−1
1 σ−12
both generate the braid group. Obviously, this should be done by imposing the right
relations. So, we define two abstract groups Bn and Bn to treat the isomorphism for-
mally. Finally, we relate the length function in the Garside generators and the length
function in the linear generators.
In section 5.1, we define a formal symbol for the generator of our abstract group
Bn. This abstract group is generated with one relation. Then, by isomorphism in The-
orem 5.1.1, we indeed see that the generator of Bn is the positive Garside generators.
In the same favour, we see that the linear generators of Bn are the linear braids.
But, a point worth noting is that the abstract group Bn has far more generators and
75
relations than the original presentation of the braid group and Bn.
Section 5.3 is the final section of this thesis. Although we don’t know much about
the length function `Artin, Theorem 5.3.2 is able to find `linear from `Garside and vice
versa with only minimal condition on the braid.
For this chapter, we will keep using the convention that W always refers to the
Weyl group of sln(C).
5.1 The Abstract Group, Bn
First, we want to build an abstract group Bn. Suppose w is an element in the Weyl
group W. The generators of Bn will be w+. Define the presentation Bn as follows:Generators : w+ | w ∈ W,Relations : x+y+ = z+ if xy = z with `(x) + `(y) = `(z) for x, y, z ∈ W,
that is,
Bn := 〈w+ | x+y+ = z+ if xy = z with `(x) + `(y) = `(z) for x, y, z ∈ W〉.
Theorem 5.1.1. The group Bn is isomorphic to the braid group Bn.
Proof. First, we define a group homomorphism Φ from Bn to Bn by
Φ : Bn → Bnσi 7→ s+
i .
It follows that Φ : σ−1i 7→ (s+
i )−1.
Moreover, we check that the relations held in the braid group Bn must also hold in
Bn. Then, for |i− j| > 1,
Φ(σiσj) = Φ(σi)Ψ(σj)
= s+i s
+j
= (sisj)+
= (sjsi)+ by the relation in W,
= s+j s
+i
= Φ(σj)Φ(σi)
= Φ(σjσi)
and for |i− j| = 1,
76
Φ(σiσjσi) = Φ(σi)Φ(σj)Φ(σi)
= s+i s
+j s
+i
= (sisj)+s+
i
= (sisjsi)+
= (sjsisj)+ by the relation in W,
= (sjsi)+s+
j
= s+j s
+i s
+j
= Φ(σj)Φ(σi)Φ(σj)
= Φ(σjσiσj),
as desired.
Next, we define a group homomorphism
Ψ : Bn → Bn
by sending w+ to the positive Garside generator σi1σi2 . . . σin = β+w if w has a reduced
expression si1 · · · sin . To check that Ψ is well defined, take any two reduced expressions
for the element w, say w1 = si1si2 · · · sin and w2 = sj1sj2 · · · sjn . From Theorem 3.1.28,
we know that they are connected by a sequence of braid-moves. Hence, their positive
lifts equal through the corresponding braid relations.
Finally, it is easy to see that Φ and Ψ are inverses of each other from the construc-
tion, so we establish an isomorphism between B and Bn.
QED
Remark 5.1.2. The generators of Bn are exactly the positive Garside generators.
Definition 5.1.3. Every element β ∈ Bn can be expressed as a product of Garside
generators:
β = (w+1 )ε1(w+
2 )ε1 . . . (w+n )εn .
If n is the minimal among all such expressions β for β, we write
`Garside(β) = n
and n is the Garside length, `Garside of β. By convention, `(e) = 0.
5.2 The Abstract Group, Bn
Let si1si2 . . . sik be a (not-necessarily reduced) expression for an element w of the Weyl
group W , with left root set RootL(w). Let A = (A+, A−) be a separated splitting of
77
RootL(w), so that
RootL(w) = A+ ∪A−,
and (A+, A−) is separated. Associated to the splitting A, we have a linear lift w of w
to an element of the braid group, as we have discussed before.
Now, consider an abstract group Bn presented as follows. The generators are wA,
where w is an expression in the Weyl group generators, and A is a separated splitting
of RootL(w). Bn has generators wA. The relations are
(R1) (si)(αi,∅)(si)(∅,αi) = e, and (si)(∅,αi)(si)(αi,∅) = e. Here α is the simple root
associated to a simple reflection si,
(R2) For |i− j| > 1, (sisj)(αi,αj,∅) = (sjsi)(αj ,αi,∅),
(R3) For |i− j| = 1, (sisjsi)(αi,αi+αj ,αj,∅) = (sjsisj)(αj ,αi+αj ,αi,∅),
(R4) uAvA′ = (uv)A∪u(A′) when A ∪ u(A′) is separated.
Note here that the concatenated word uv is an element of W. Also, if A = (A+, A−)
and A′ = (A′+, A′−), then
A ∪ w(A′) := (A+ ∪ w(A′+
), A− ∪ w(A′−
)).
Let S∗ be the free monoid on the generating set S of W. Therefore,
Bn := 〈wA, w ∈ S∗ and A is a splitting of RootL(w) | R1, R2, R3, R4〉.
So, Bn has more generators (and more relations) in its presentation than Bn does.
Nonetheless we have the following theorem.
Theorem 5.2.1. The group Bn is isomorphic to the braid group Bn .
Proof. The idea of the proof is to define a group homomorphism
Φ : Bn −→ Bn
by sending wA to the linear lift βw whose left root set splitting is A. The claim is Φ is
an isomorphism.
First, we show that Φ is a well-defined group homomorphism. Under the map Φ,
(si)(αi,∅)(si)(∅,αi) 7→ σiσ−1i = e and
(si)(∅,αi)(si)(αi,∅) 7→ σ−1i σi = e.
So, it is sent to identity in Bn under the homomorphism. Moreover, for |i− j| > 1,
Φ((sisj)(αi,αj,∅)) = σiσj = σjσi = Φ((sjsi)(αj ,αi,∅)),
78
and for |i− j| = 1,
Φ((sisjsi)(αi,αi+αj ,αj,∅)) = σiσjσi = σjσiσj = (sjsisj)(αj ,αi+αj ,αi,∅).
Suppose u = si1si2 . . . sin with a separated splitting A and v = sj1sj2 . . . sjm with a
separated splitting A′. Then, there exists ε1, ε2, . . . , εn, η1, η2, . . . , ηm ∈ ±1,
Φ(uAvA′) = σε1i1 σε2i2. . . σεnin σ
η1j1ση2j2 . . . σ
ηmjm
= Φ((uv)A∪u(A′))
since if A ∪ u(A′) is a separated splitting implying the image of (uv)A∪u(A′) under Φ
must be linear, then its subexpressions must have a separated splitting of root set by
Proposition 4.3.12.
Next, we want to define its inverse
Ψ : Bn → Bn
sending σi to (si)(αi,∅). Then, we need to check that Ψ is well-defined. For |i− j| > 1,
Ψ(σiσj) = Ψ(σi)Ψ(σj)
= (si)(αi,∅)(sj)(αj,∅)
= (sisj)(αi,αj,∅)
= (sjsi)(αj ,αi,∅)
= (sj)(αj,∅)(si)(αi,∅)
= Ψ(σj)Ψ(σi)
= Ψ(σjσi)
and for |i− j| = 1,
Ψ(σiσjσi) = Ψ(σi)Ψ(σj)Ψ(σi)
= (si)(αi,∅)(sj)(αj,∅)(si)(αi,∅)
= (sisj)(αi,αi+αj,∅)(si)(αi,∅)
= (sisjsi)(αi,αi+αj ,αj,∅)
= (sjsisj)(αj ,αi+αj ,αi,∅)
= (sjsi)(αj ,αi+αj,∅)(sj)(αj,∅)
= (sj)(αj,∅)(si)(αi,∅)(sj)(αj,∅)
= Ψ(σj)Ψ(σi)Ψ(σj)
= Ψ(σjσiσj),
as desired.
The fact that they are inverses of each other is clear from the construction.
QED
79
Definition 5.2.2. Every element β ∈ Bn can be expressed as a product of linear gen-
erators:
β = l1 l2 . . . ln.
If n is the minimal among all such expressions β for β, we write
`linear(β) = n
and n is the linear length, `linear of β. By convention, `linear(e) := 0.
Recall from the previous chapter that a left descent set DL(β) for a braid is defined
as
DL(β) := σεj | β has a minimal length expression β = σεjσε1i1σε2i2 . . . σ
εmim.
and a right descent set DR(β) for a braid is defined as
DR(β) := σεk | β has a minimal length expression β = σε1i1 σε2i2. . . σεmim σ
εkk . .
5.3 Relation between `linear and `Garside
The next proposition is concerning an arbitrary braid.
Proposition 5.3.1. Any braid β has an expression
β1+ β2
− = β3− β4
+,
for some positive expressions β1+, β3
− and some negative expressions β2−, β4
+.
Proof. Note that we can decompose β = (w+1 )ε1(w+
2 )ε1 . . . (w+4 )ε1 as a product of Gar-
side generators.
By repeatedly application of Proposition 4.3.19, we can get
β = β1+β2
− = β3−β4
+
for some positive expressions β1+, β4
+ and some negative expressions β2−, β3
−.
QED
Theorem 5.3.2. Write an expression β = β1+β2
− as a product of a positive expression
and a negative expression with DR(β1) ∩ DL(β2) = ∅ such that there is no internal
cancellation of terms in the expression for the braid β.
Then,
`linear(β) = max(`Garside(β1), `Garside(β2)).
80
Proof. First, we will show that
`linear(β) ≤ max(`Garside(β1), `Garside(β2)).
Suppose `Garside(β1) = m and `Garside(β2) = n. Without loss of generality, assume m ≤n. Then, we can write, for some Garside generators x+
1 , x+2 , . . . , x
+m,
(y+1 )−1, (y+
2 )−1, . . . , (y+n )−1,
β = x+1 x
+2 . . . x
+m(y+
1 )−1(y+2 )−1 . . . (y+
n )−1.
By repeatedly application of Proposition 4.3.19, we get
β = x+1 (u+
1 )−1v+2 (u+
2 )−1 . . . v+m(u+
m)−1(y+m+1)−1 . . . (y+
n )−1,
for some Garside generator (u+1 )−1, . . . , (u+
m)−1, v+2 , . . . , v
+m. From Theorem 4.3.25 again,
we see that the first m pairs of Garside generators are linear. Since the remaining
Garside generators are clearly linear, we have
`linear(β) ≤ n.
On the other hand, if m > n, we see that
`linear(β) ≤ m
using a similar argument. Therefore, `linear(β) ≤ max(`Garside(β1), `Garside(β2)).
Next, we will show that
`linear(β) ≥ max(`Garside(β1), `Garside(β2)).
Suppose `linear(β) = k, so we can write, for some linear generators l1, l2, . . . , lk,
β = l1 l2 . . . lk.
By Theorem 4.3.25, we can decompose each linear generator into a product of positive
and negative Garside generators, that is
β = x+1 (y+
1 )−1x+2 (y+
2 )−1 . . . x+k (y+
k )−1.
Again, by Proposition 4.3.19, we obtain
β = x+1 v
+2 . . . v
+k (u+
1 )−1(u+2 )−1 . . . (u+
k )−1
such that it is a product of some positive and negative Garside generators. From here,
it is clear that `Garside(β1) ≤ k and `Garside(β2
) ≤ k. Hence,
max(`Garside(β1), `Garside(β2)) ≤ `linear(β).
Together with the previous result, we have
`linear(β) = max(`Garside(β1), `Garside(β2)).
QED
81
82
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