← Matrix Lie-Groups → Topological Groups

What should you be acquainted with? 1. Linear Algebra, in particular inner product spaces both over the real and the complex numbers. 2. Very basics in Group Theory and Complex Analysis. This chapter is essentially taken from Brian Hall, Lie Groups, Lie Algebras, and Representations, Chapter 6.

Representations of $s u (3)$ and $S U (3)$

Weights and Roots
Heighest Weight Theorem
Some Examples
The Weyl Group

Weights and Roots

Complexifying su(n)

E^{j k}

denote the standard basis of

M (n, C)

, then the matrices

H_{1} : = E^{11} - E^{22}

\dots

H_{n - 1} : = E^{n - 1, n - 1} - E^{n n}

j < k

A_{j k} : = E^{j k} - E^{k j}

B_{j k} : = i E^{j k} + i E^{k j}

form a basis of

s u (n)

. Hence for any

U \in S U (n)

there are real numbers

h_{j}, a_{j k}, b_{j k}

U = \sum_{j < n} h_{j} H_{j} + \sum_{j < k} a_{j k} A_{j k} + b_{j k} B_{j k}

The complexification of

s u (n)

is the space of these linear combinations but the reals

h_{j}, a_{j k}, b_{j k}

replaced with complex numbers. As we've seen (exam) the complexification coincides with the complex Lie-algebra

s l (n, C)

of traceless complex

n

by

n

matrices.

H_{j} / \sqrt{2}

A_{j k} / \sqrt{2}

B_{j k} / \sqrt{2}

form an orthonormal basis of

s u (n)

with respect to its canonical euclidean product:

⟨ A, B ⟩ = t r (A B^{*})

and the extension of this product to

s l (n, C)

is also given by

⟨ A, B ⟩ = t r (A B^{*})

n = 3

we choose the following eight matrices as basis for

s l (3, C)

H_{1} : = (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \end{array}), X_{1} : = (\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}) = E^{12}, Y_{1} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}) = E^{21}

H_{2} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{array}), X_{2} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}) = E^{23}, Y_{2} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{array}) = E^{32}

X_{3} : = (\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}) = E^{13}, Y_{3} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}) = E^{31}

X_{j}, Y_{j} \notin s u (3)

s p a n (H_{1}, X_{1}, Y_{1})

s p a n (H_{2}, X_{2}, Y_{2})

are obviously subalgebras and isomorphic to

s l (2, C)

\forall j \in {1, 2} [H_{j}, X_{j}] = 2 X_{j}, [H_{j}, Y_{j}] = - 2 Y_{j}, [X_{j}, Y_{j}] = H_{j}

Also, the subspace

s p a n (H_{1}, H_{2})

is the (complexified) Lie-algebra of the torus

T^{2}

, i.e. the traceless diagonal matrices; this subalgebra is commutative and thus

[H_{1}, H_{2}] = 0

. Moreover we have the following commutation relations:

\begin{array}{rcl} [H_{1}, X_{2}] = - X_{2}, [H_{1}, Y_{2}] = Y_{2}, [H_{1}, X_{3}] = X_{3}, [H_{1}, Y_{3}] = - Y_{3} \\ [H_{2}, X_{1}] = - X_{1}, [H_{2}, Y_{1}] = Y_{1}, [H_{2}, X_{3}] = X_{3}, [H_{2}, Y_{3}] = - Y_{3} \end{array}

X_{1} X_{2} = X_{3}

Y_{2} Y_{1} = Y_{3}

and all other products

X_{j} X_{k}

Y_{j} Y_{k}

vanish. Thus we have

X_{3} = [X_{1}, X_{2}]

Y_{3} = - [Y_{1}, Y_{2}]

H_{3} : = - H_{1} - H_{2}

[H_{3}, X_{3}] = - 2 X_{3}, [H_{3}, Y_{3}] = 2 Y_{3}, [X_{3}, Y_{3}] = - H_{3}

s p a n (H_{3}, X_{3}, Y_{3})

is a subalgebra isomorphic to

s l (2, C)

Weights

The classification of the irreducible representations of

S U (2)

heavily relied on the fact that both

X_{1}

Y_{1}

are ladder operators for

H_{1}

. This is still true for

S U (3)

but there are several additional commutation relations, one of which is

[H_{1}, H_{2}] = 0

, which implies that given any representation

ψ : s l (n, C) \to H o m (E)

in a finite dimensional complex vector-space

E

, the linear mappings

ψ (H_{1})

ψ (H_{2})

have at least one joint eigen-vector - if both

ψ (H_{1})

ψ (H_{2})

are diagonalizable, then

E

has a basis of joint eigen-vectors. Indeed, we can always assume that these operators are jointly diagonalizable, because

s u (n)

is the Lie-algebra of the compact group

S U (n)

and hence for any representation

ψ : s l (n, C) \to H o m (E)

there is a euclidean product on

E

such that for all

X \in s u (n)

ψ (X)^{*} = - ψ (X)

(cf section). This implies that

ψ

is the direct sum of irreducible representations. Indeed if

F

is invariant, then so is

F^{⊥}

, because for all

y \in F^{⊥}

and all

x \in F

:

⟨ ψ (X) y, x ⟩ = ⟨ y, ψ (X)^{*} x ⟩ = - ⟨ y, ψ (X) x ⟩ = 0 .

From this it follows (cf. proposition) that

ψ

is the direct sum of irreducible representations.
Let

A

be the space generated by

ψ (H_{1})

and

ψ (H_{2})

, then by subsection a weight of

A

can be identified with a pair

λ : = (λ (H_{1}), λ (H_{2})) \in C^{2}

, such that there exists a common eigen-vector

x \in E ∖ {0}

:

ψ (H_{1}) x = λ (H_{1}) x

and

ψ (H_{2}) x = λ (H_{2}) x

.

λ

will be called a weight of the representation

ψ

and

x

is called a weight vector corresponding to the weight

λ

. The space of all weight vectors for the weight

λ

is called the weight space of

λ

and the dimension of the weight space of

λ

is called the multiplicity of the weight

λ

. By restricting

ψ

to the subalgebra generated by

H_{1}, X_{1}, Y_{1}

and

H_{2}, X_{2}, Y_{2}

respectivley, we know from proposition that both components of a weight must be integers.
Let us calculate the weights of the standard representation of

s l (3, C)

and the weights of its dual representation: obviously the canonical basis

e_{1}, e_{2}, e_{3}

of

C^{3}

are weight vectors with weights:

(1, 0)

,

(- 1, 1)

and

(0, - 1)

. Since the dual representation is the negative of the transpose,

e_{1}, e_{2}

and

e_{3}

are also the weight vectors of the dual with weights:

(- 1, 0)

,

(1, - 1)

,

(0, 1)

. Thus the weights of the dual of the standard representation are the negatives of the weights of the standard representation. This is actually true for any representation if we assume that

ψ (H_{1})

and

ψ (H_{2})

are jointly diagonalizable, which is no loss of generality. Under this condition the weights of the dual representation

ψ^{d}

of

ψ : s l (3, C) \to H o m (E)

are the negatives of the weights of

ψ

: Choose a basis

e_{1}, \dots, e_{n}

of

E

such that both

ψ (H_{1})

and

ψ (H_{2})

are diagonal with respect to this basis and

ψ (H_{j}) e_{k} = m_{j k} e_{k}

. Then the matrix of

ψ^{d} (H_{j})

with respect to the dual basis

e_{1}^{*}, \dots, e_{n}^{*}

is the negative of the transpose of the matrix of

ψ (H_{j})

with respect to the basis

e_{1}, \dots, e_{n}

(cf. section). Hence

ψ^{d} (H_{j}) e_{k}^{*} = - m_{j k} e_{k}^{*}

.

Roots

A pair

r : = (r (H_{1}), r (H_{2})) \in C^{2} ∖ {(0, 0)}

is said to be a root if for some

Z \in s l (3, C)

[H_{1}, Z] = r (H_{1}) Z

[H_{2}, Z] = r (H_{2}) Z

Z

is a ladder operator for both

H_{1}

H_{2}

and it doesn't commute with both.

Z \in s l (3, C)

is called a root vector . Equivalently we may say that

Z

is an eigen-vector for both

a d (H_{1})

a d (H_{2})

and the corresponding eigen-values are

r (H_{1})

r (H_{2})

and this in turn means:

Roots are the non zero weights of the adjoint representation.

The above commutation relations give us the following roots

\begin{array}{rc} root & root vector \\ r_{1} = (2, - 1) & X_{1} \\ r_{2} = (- 1, 2) & X_{2} \\ r_{3} = r_{1} + r_{2} = (1, 1) & X_{3} \\ - r_{1} = (- 2, 1) & Y_{1} \\ - r_{2} = (1, - 2) & Y_{2} \\ - r_{3} = (- 1, - 1) & Y_{3} \end{array}

i.e. all vectors

X_{j}, Y_{j}

,

j = 1, 2, 3

are root vectors. The following is a straightforward generalization of example:

r

be a root with corresponding root vector

Z

ψ : s l (3, C) \to H o m (E)

is a finite dimensional representation,

λ

ψ

x \neq 0

a weight vector. Then for

j = 1, 2, 3

ψ (H_{j}) ψ (Z) x = (λ (H_{j}) + r (H_{j})) ψ (Z) x .

Therefore, either

ψ (Z) x = 0

or it's a new weight vector with weight

λ + r

Proof :

ψ

is an algebra homomorphism and

[H_{j}, Z] = r (H_{j}) Z

, it follows that

\begin{array}{rcl} ψ (H_{j}) ψ (Z) x & = & [ψ (H_{j}), ψ (Z)] x + ψ (Z) ψ (H_{j}) x \\ = & r (H_{j}) ψ (Z) x + λ (H_{j}) ψ (Z) x = (λ (H_{j}) + r (H_{j})) ψ (Z) x \end{array}

qed

Just like eigen-vectors for pairwise distict eigen-values are linearly independent, root vectors for pairwise distict roots are linearly independent:

x_{1}, \dots, x_{n}

are weight vectors for mutually distinct weights

λ_{1}, \dots, λ_{n}

x_{1} + \dots + x_{n}

is a weight vector for the weight

λ

, then for some

l

λ = λ_{l}

x = x_{l}

k \neq l

x_{k} = 0

Proof :

x_{1} + \dots + x_{n} = 0

n \geq 2

. We procede by induction on

n

to prove that this assumption implies that all vectors

x_{1}, \dots, x_{n}

0

n = 1

the assertion is obvious. So assume

n \geq 2

λ_{1} \neq λ_{2}

, there is some

j

λ_{1} (H_{j}) \neq λ_{2} (H_{j})

0 = (ψ (H_{j}) - λ_{1} (H_{j})) \sum_{k} x_{k} = \sum_{k \geq 2} (λ_{k} (H_{j}) - λ_{1} (H_{j})) x_{k} .

By induction hypothesis all terms of the last sum vanish, in particular

x_{2} = 0

x_{1} + x_{3} + \dots + x_{n} = 0

which again by induction hypothesis implies that all vectors

x_{k}

x : = x_{1} + \dots + x_{n}

is a non zero weight vector with weight

λ

j

0 = (ψ (H_{j}) - λ (H_{j})) x = \sum_{k} (λ_{k} (H_{j}) - λ (H_{j})) x_{k}

and by 1. it follows that for all

k

(λ_{k} (H_{j}) - λ (H_{j})) x_{k} = 0

x \neq 0

, there is some

l

x_{l} \neq 0

and thus for all

j

λ_{l} (H_{j}) = λ (H_{j})

λ = λ_{l}

k \neq l

then there is some

j

λ_{k} (H_{j}) \neq λ_{l} (H_{j})

x_{k} = 0

qed

ψ : s l (3, C) \to H o m (E)

is a finite dimensional irreducible complex representation. Then

E

is the direct sum of its weight spaces.

Proof :

This is obvious if

ψ (H_{j})

are diagonalizable, but we actually don't need this assumption. We already observed that

ψ (H_{1})

ψ (H_{2})

have at least one common complex eigen-vector. Now let

W \subseteq E

be the subspace of all weight spaces, then

W \neq {0}

X_{j}

Y_{j}

j = 1, 2, 3

, are root vectors, the operators

ψ (X_{j})

ψ (Y_{j})

W

into itself by lemma. Obviously,

W

is invariant under

ψ (H_{j})

und thus it's invariant under all operators

ψ (X)

,

X \in s l (3, C)

. By irreducibility

E

is the sum of its weight spaces and by lemma the sum is direct.

qed

Heighest Weight Theorem

Similar to the $s u (2)$ case we may start with some weight $λ$ and some weight vector $x$ and apply the operators $ψ (X_{j})$ and $ψ (Y_{j})$ in any order to get some new weight vectors according to lemma. However, since $E$ is finite dimensional we must eventually get the null vector. Is there a way to start with some "highest" weight and work down to get all the others? Yes, this can be done, but it's not as easy as in the $s u (2)$ case: First let us single out the roots $r_{1} = (2, - 1)$ and $r_{2} = (- 1, 2)$ ; it's easily checked that all the other roots are linear combination of these two: e.g. the root corresponding to the root vector $X_{3}$ is just the sum of $r_{1}$ and $r_{2}$ .

ψ : s l (3, C) \to H o m (E)

be a representation. On the set of all weights we define an order by putting:

λ_{1} ⪯ λ_{2}

if the difference

λ_{2} - λ_{1}

is a non negative (i.e.

(a_{1}, a_{2}) \neq (0, 0)

) real linear combination

a_{1} r_{1} + a_{2} r_{2}

r_{1}

r_{2}

λ_{2}

λ_{1}

ψ : s l (3) \to H o m (E)

is a representation and

λ

ψ

λ

is said to be a highest weight if for all weights

μ

ψ

μ ⪯ λ

In the picture below the weight (0,0) is higher than any weight in the blue sector!

highest weight

(1, 0)

for the standard representation of

s l (3, C)

is higher than the weight

(0, 0)

(1, 0) - (0, 0) = \frac{2}{3} r_{1} + \frac{1}{3} r_{2}

(1, 0) - (- 1, 1) = (2, 1)

(1, 0) - (0, - 1) = (1, 1) = r_{1} + r_{2}

(1, 0)

is a highest weight for the standard representation. Similarly

(0, 1)

is a highest weight for the dual representation.

A representation can only have one highest weight.

A highest weight is of course a maximal element (with respect to

⪯

) but in general it's not the other way round; even if the set of weights is finite - which is the case if the representation is finite dimensional - a highest weight may not exist! So let us start with a maximal weight

λ

of a representation

ψ : s l (3, C) \to H o m (E)

with corresponding weight vector

x

ψ (X_{1}) x = ψ (X_{2}) x = 0

, for otherwise

λ

wouldn't be maximal. Moreover, since the root

(1, 1)

for the root vector

X_{3}

can be written as

(1, 1) = r_{1} + r_{2}

, we must also have

ψ (X_{3}) x = 0

ψ : s l (3, C) \to H o m (E)

is a highest weight cyclic representation if the following holds:

There exists a weight vector $x \in E ∖ {0}$ with weight $λ$ .
For all $j = 1, 2, 3$ : $ψ (X_{j}) x = 0$ .
$x$ is a cyclic vector, i.e. the space generated by $ψ (X) x$ , $X \in s l (3, C)$ is all of $E$ .

Restricting a highest weight cyclic representation to the subalgebras generated by

H_{1}, X_{1}, Y_{1}

H_{2}, X_{2}, Y_{2}

respectivley, we get representations of

s l (2, C)

ψ (X_{1}) = ψ (X_{2}) = 0

and thus by proposition we infer that both

λ (H_{1})

and

λ (H_{2})

must be non negative integers. Also, if a weight vector for a maximal weight is cyclic, then

ψ

is a highest weight cyclic representation and both components of the weight are non negative integers. But more is true and this also explains why a highest weight cyclic representation is called highest weight representation:

ψ : s l (3, C) \to H o m (E)

be a highest weight cyclic representation with weight

λ

and weight vector

x

λ

is the highest weight of

ψ

and the corresponding weight space is one dimensional.

Proof :

First we proof that the space

W

ψ (Y_{j_{1}}) \dots ψ (Y_{j_{m}}) x

is invariant: By the reordering lemma it suffices to prove that all elements of the form

ψ (Y_{j_{1}}) \dots ψ (Y_{j_{m}}) ψ (H_{1})^{k_{1}} ψ (H_{2})^{k_{2}} ψ (X_{1})^{l_{1}} ψ (X_{2})^{l_{2}} ψ (X_{3})^{l_{3}} x

live in

W

, but

ψ (X_{j}) x = 0

and

ψ (H_{j}) x = λ (H_{j}) x

and thus

W

is

ψ

-invariant by its definition. Since

x

is cyclic, we conclude:

W = E

. Now suppose

y

is any vector in

E = W

of the form

y = ψ (Y_{j_{1}}) \dots ψ (Y_{j_{m}}) x

; since

Y_{1}, Y_{2}, Y_{3}

are root vectors with roots

- r_{1}, - r_{2}, - r_{1} - r_{2}

,

y

is a weight vector and by lemma its weights are stricly lower than

λ

unless

y = x

. Therefore

W

has a basis

x, y_{1}, \dots, y_{m}

of weight vectors and each weight of

y_{j}

is stricly lower than

λ

. By lemma we are done.

qed

Now assume moreover that

ψ

is irreducible, then any vector in

E ∖ {0}

is cyclic and thus any non zero weight vector for a maximal weight is cyclic and satisfies

ψ (X_{j}) = 0

, i.e. every irreducible representation is a highest weight cyclic representation. Also the converse holds

ψ : s l (3, C) \to H o m (E)

be a finite dimensional representation. Then

ψ

is irreducible if and only if

ψ

is a heighest weight cyclic representation.

Proof :

E

is finite dimensional it decomposes into subspaces

E = ⨁ E_{j}

ψ : s l (3, C) \to H o m (E_{j})

is irreducible. Each of these spaces

E_{j}

in turn decomposes by proposition into its weight spaces and by lemma the weight vector

x

must lie in one of these spaces and consequently in one of the spaces

E_{j}

. Since

x

is cyclic and

E_{j}

is invariant, we must have

E = E_{j}

.

qed

So far we have established that an irreducible representation is the same as a highest weight cyclic representation, moreover the weight space of the highest weight of this representations is one dimensional and the components of the heighest weight are non negative. Our next goal is to verify the following

Two irreducible representations with the same highest weight

λ

are equivalent.

Proof :

ψ : s l (3, C) \to H o m (E)

φ : s l (3, C) \to H o m (F)

be irreducible representations with the highest weight

λ

u \in E

v \in F

be weight vectors with this weight. Then

(u, v)

is a weight vector of the representation

π : s l (3, C) \to H o m (E \times F)

π (x, y) = (ψ (x), φ (y))

W

be the subspace generated by

(u, v)

π : s l (3, C) \to H o m (W)

is a highest weight cyclic representation, by proposition it's irreducible. Further, the projections

P : W \to E

,

(x, y) \mapsto x

and

Q : W \to F

,

(x, y) \mapsto y

are intertwining operators, i.e.

P π (x, y) = ψ (x) = ψ (P (x, y))

and

Q π (x, y) = φ (y) = φ (Q (x, y))

and since

P (u, v) = u \neq 0

and

Q (u, v) = v \neq 0

, both

P : W \to E

and

Q : W \to F

must be isomorphisms by Schur's lemma for Lie algebras, proving that both irreducible representations

ψ : s l (3, C) \to H o m (E)

and

φ : s l (3, C) \to H o m (F)

are equivalent to the irreducible representation

π : s l (3, C) \to H o m (W)

.

qed

Our last step is the construction of irreducible representations

For every pair of non negative integers

(m_{1}, m_{2})

there is an irreducible representation of

s l (3, C)

, whose highest weight

λ

λ (H_{j}) = m_{j}

Proof :

The trivial representation

X \mapsto 0

has highest weight

(0, 0)

, the standard representation

ψ

has highest weight

(1, 0)

φ

has highest weight

(0, 1)

E = F = C^{3}

u \in E

v \in F

be weight vectors for these two representations with the highest weight

(1, 0)

(0, 1)

π

be the following representation

π : s l (3, C) \to H o m (E \otimes \dots \otimes E \otimes F \otimes \dots \otimes F)

m_{1}

-fold tensor products of

E

m_{2}

-fold tensor products of

F

π

is defined by the sum of

m_{1} + m_{2}

\begin{array}{rcl} π (X) : & = & ψ (X) \otimes 1 \otimes \dots \otimes 1 + \dots + 1 \otimes \dots \otimes 1 \otimes ψ (X) \otimes 1 \otimes \dots \otimes 1 \\ + 1 \otimes \dots \otimes 1 \otimes φ (X) \otimes 1 \otimes \dots \otimes 1 + \dots + 1 \otimes \dots \otimes 1 \otimes φ (X) \end{array}

w : = u \otimes \dots \otimes u \otimes v \otimes \dots \otimes v

. Then it follows that

π (H_{j}) w = m_{j} w and π (X_{j}) w = 0 .

π

restricted to the space

W

invariant under

π (Y_{j})

j = 1, 2, 3

, is an irreducible representation with highest weight

(m_{1}, m_{2})

. In order to get

W

we observe that

Y_{3} = [Y_{2}, Y_{1}]

and thus we only need to apply the operators

π (Y_{j})

j = 1, 2

, in every order to

w

qed

u

v

are weight vectors for

ψ

φ

λ

μ

u \otimes v

is a weight vector for

π : = ψ \otimes 1 + 1 \otimes φ

λ + μ

π (H_{j}) u \otimes u = ψ (H_{j}) u \otimes v + u \otimes φ (H_{j}) v = (λ (H_{j}) + μ (H_{j})) u \otimes v

Some Examples

$1$ -, $3$ - and $\bar{3}$ - representation, quark notation

These just denote the trivial, the standard and its dual representation. In particle physics the weight vectors

e_{1}, e_{2}, e_{3}

of the standard representation, i.e. the common eigen-vectors of

H_{1}

H_{2}

u, d, s

- up, down and strange quark. Hence

u, d, s

form an orthonormal basis of the

3

-representation; accordingly, the antiparticles

\bar{u}, \bar{d}, \bar{s}

form an orthonormal basis of the

\bar{3}

-representation. Mathematically they are simply the weight vectors of the standard representation and its dual. In physics particles made of any number of quarks are called hadrons , also the matrices

I_{3} : = H_{1} / 2

Y : = (H_{1} + 2 H_{2}) / 3

Q : = \frac{1}{2} Y + I_{3}

are used. Their eigen-values corresponding to the eigen-vectors

u, d, s

are called the isospin , the hypercharge and the (electric) charge , respectively, cf. wikipedia. We will see below (cf. section) that these are in a way dual operators to

H_{1}

and

H_{2}

.

I_{3}

Y

Q

(\begin{array}{ccc} 1 / 2 & 0 & 0 \\ 0 & - 1 / 2 & 0 \\ 0 & 0 & 0 \end{array}), (\begin{array}{ccc} 1 / 3 & 0 & 0 \\ 0 & 1 / 3 & 0 \\ 0 & 0 & - 2 / 3 \end{array}), (\begin{array}{ccc} 2 / 3 & 0 & 0 \\ 0 & - 1 / 3 & 0 \\ 0 & 0 & - 1 / 3 \end{array})

Hence the isospins, the hypercharges and the charges of

u, d, s

(- 1 / 2, 1 / 2, 0)

(1 / 3, 1 / 3, - 2 / 3)

(2 / 3, - 1 / 3, - 1 / 3)

. The diagonal matrix

B : = d i a g {1 / 3, 1 / 3, 1 / 3}

is called baryon number .

Heighest Weight $(2, 0)$ - the $6$ -representation

m_{1} = 2

m_{2} = 0

π (X) = X \otimes 1 + 1 \otimes X

and the highest weight vector is

e_{1} \otimes e_{1}

. We have to find the smallest space

W

w : = e_{1} \otimes e_{1}

and invariant under the action of the operators

π (Y_{1}) = Y_{1} \otimes 1 + 1 \otimes Y_{1} and π (Y_{2}) = Y_{2} \otimes 1 + 1 \otimes Y_{2}

applied in any order.

\begin{array}{rcl} π (Y_{1}) (e_{1} \otimes e_{1}) & = & e_{2} \otimes e_{1} + e_{1} \otimes e_{2} \\ π (Y_{2}) (e_{1} \otimes e_{1}) & = & 0 \\ π (Y_{1}) (e_{2} \otimes e_{1} + e_{1} \otimes e_{2}) & = & 2 e_{2} \otimes e_{2} \\ π (Y_{2}) (e_{2} \otimes e_{1} + e_{1} \otimes e_{2}) & = & e_{3} \otimes e_{1} + e_{1} \otimes e_{3} \\ π (Y_{1}) (e_{2} \otimes e_{2}) & = & 0 \\ π (Y_{2}) (e_{2} \otimes e_{2}) & = & e_{3} \otimes e_{2} + e_{2} \otimes e_{3} \\ π (Y_{1}) (e_{3} \otimes e_{2} + e_{2} \otimes e_{3}) & = & 0 \\ π (Y_{2}) (e_{3} \otimes e_{2} + e_{2} \otimes e_{3}) & = & 2 e_{3} \otimes e_{3} \\ π (Y_{1}) (e_{3} \otimes e_{3}) & = & 0 \\ π (Y_{2}) (e_{3} \otimes e_{3}) & = & 0 \end{array}

Thus the irreducible representation with heighest weight

(2, 0)

is of dimension

6

- hence it's called the

6

-representation - with basis:

e_{1} \otimes e_{1}, e_{2} \otimes e_{2}, e_{3} \otimes e_{3}, e_{1} \otimes e_{2} + e_{2} \otimes e_{1}, e_{2} \otimes e_{3} + e_{3} \otimes e_{2}, e_{3} \otimes e_{1} + e_{1} \otimes e_{3} .

In particle physics the vectors

e_{1} \otimes e_{1}, \dots, e_{3} \otimes e_{3}

are designated as

u u, \dots, s s

, which flag particles composed of two quarks. Hence we have the following orthonormal basis in "quark"-notation:

u u, d d, s s, \frac{u d + d u}{\sqrt{2}}, \frac{d s + s d}{\sqrt{2}}, \frac{u s + s u}{\sqrt{2}} .

u, d, s

(1, 0), (- 1, 1), (0, - 1)

, these vectors have the weights:

(1, 0) + (1, 0) = (2, 0), (- 2, 2), (0, - 2), (0, 1), (- 1, 0), (1, 0) + (0, - 1) = (1, - 1) .

The following diagram depicts the

6

-representation

π

π (Y_{1})

π (Y_{2})

π (X_{1})

π (X_{2})

map each basis vector, represented by a circle enclosing its weight, to a multiple of another basis vector. This multiple is attached to the correspondingly colored arrow, i.e. the blue, cyan, red and green arrow indicates the action of

π (Y_{1})

π (Y_{2})

π (X_{1})

π (X_{2})

. If there is e.g. no outgoing blue arrow from one of the circles, then the corresponding basis vector is mapped by

π (Y_{1})

to the null vector!

6-representation

Heighest Weight $(1, 1)$ - the $8$ -representation

This time we have

π (X) = X \otimes 1 + 1 \otimes \bar{X}

, where the matrices

{\bar{H}}_{j}

{\bar{X}}_{j}

{\bar{Y}}_{j}

{\bar{H}}_{1} : = (\begin{array}{ccc} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}), {\bar{X}}_{1} : = (\begin{array}{ccc} 0 & 0 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}), {\bar{Y}}_{1} : = (\begin{array}{ccc} 0 & - 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array})

{\bar{H}}_{2} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array}), {\bar{X}}_{2} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & - 1 & 0 \end{array}), {\bar{Y}}_{2} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 0 & 0 \end{array})

{\bar{X}}_{3} : = (\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ - 1 & 0 & 0 \end{array}), {\bar{Y}}_{3} : = (\begin{array}{ccc} 0 & 0 & - 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array})

e_{1}, e_{2}, e_{3}

be the standard basis of

C^{3}

e_{1}

is the weight vector for the standard representation with the highest weight

(1, 0)

{\bar{e}}_{3}

is the weight vector for the dual representation with the highest weight

(0, 1)

. We are left to identify the smallest space

W

w : = e_{1} \otimes {\bar{e}}_{3}

and invariant under

π (Y_{1}) = Y_{1} \otimes 1 + 1 \otimes {\bar{Y}}_{1} and π (Y_{2}) = Y_{2} \otimes 1 + 1 \otimes {\bar{Y}}_{2} .

We start out with the vector

e_{1} \otimes {\bar{e}}_{3}

\begin{array}{rcl} Y_{1} e_{1} \otimes {\bar{e}}_{3} + e_{1} \otimes {\bar{Y}}_{1} {\bar{e}}_{3} & = & e_{2} \otimes {\bar{e}}_{3} \\ Y_{2} e_{1} \otimes {\bar{e}}_{3} + e_{1} \otimes {\bar{Y}}_{2} {\bar{e}}_{3} & = & - e_{1} \otimes {\bar{e}}_{2} \end{array}

with weights: Thus we have to apply the operators to

e_{2} \otimes {\bar{e}}_{3}

e_{1} \otimes {\bar{e}}_{2}

\begin{array}{rcl} Y_{1} e_{2} \otimes {\bar{e}}_{3} + e_{2} \otimes {\bar{Y}}_{1} {\bar{e}}_{3} & = & 0 \\ Y_{2} e_{2} \otimes {\bar{e}}_{3} + e_{2} \otimes {\bar{Y}}_{2} {\bar{e}}_{3} & = & e_{3} \otimes {\bar{e}}_{3} - e_{2} \otimes {\bar{e}}_{2} \\ Y_{1} e_{1} \otimes {\bar{e}}_{2} + e_{1} \otimes {\bar{Y}}_{1} {\bar{e}}_{2} & = & e_{2} \otimes {\bar{e}}_{2} - e_{1} \otimes {\bar{e}}_{1} \\ Y_{2} e_{1} \otimes {\bar{e}}_{2} + e_{1} \otimes {\bar{Y}}_{2} {\bar{e}}_{2} & = & 0 \end{array}

This gives another two vectors:

e_{3} \otimes {\bar{e}}_{3} - e_{2} \otimes {\bar{e}}_{2}

e_{2} \otimes {\bar{e}}_{2} - e_{1} \otimes {\bar{e}}_{1}

\begin{array}{rcl} π (Y_{1}) (e_{3} \otimes {\bar{e}}_{3} - e_{2} \otimes {\bar{e}}_{2}) & = & e_{2} \otimes {\bar{e}}_{1} \\ π (Y_{2}) (e_{3} \otimes {\bar{e}}_{3} - e_{2} \otimes {\bar{e}}_{2}) & = & - 2 e_{3} \otimes {\bar{e}}_{2} \\ π (Y_{1}) (e_{2} \otimes {\bar{e}}_{2} - e_{1} \otimes {\bar{e}}_{1}) & = & - 2 e_{2} \otimes {\bar{e}}_{1} \\ π (Y_{2}) (e_{2} \otimes {\bar{e}}_{2} - e_{1} \otimes {\bar{e}}_{1}) & = & e_{3} \otimes {\bar{e}}_{2} \end{array}

which gives the two vectors:

e_{2} \otimes {\bar{e}}_{1}

e_{3} \otimes {\bar{e}}_{2}

\begin{array}{rcl} π (Y_{1}) (e_{2} \otimes {\bar{e}}_{1}) & = & 0 \\ π (Y_{2}) (e_{2} \otimes {\bar{e}}_{1}) & = & e_{3} \otimes {\bar{e}}_{1} \\ π (Y_{1}) (e_{3} \otimes {\bar{e}}_{2}) & = & - e_{3} \otimes {\bar{e}}_{1} \\ π (Y_{2}) (e_{3} \otimes {\bar{e}}_{2}) & = & 0 \end{array}

\begin{array}{rcl} π (Y_{1}) (e_{3} \otimes {\bar{e}}_{1}) & = & 0 \\ π (Y_{2}) (e_{3} \otimes {\bar{e}}_{1}) & = & 0 \end{array}

W

is generated by the eight vectors and therefore it's called the

8

-representation:

e_{1} \otimes {\bar{e}}_{2}, e_{1} \otimes {\bar{e}}_{3}, e_{2} \otimes {\bar{e}}_{1}, e_{2} \otimes {\bar{e}}_{3}, e_{3} \otimes {\bar{e}}_{1}, e_{3} \otimes {\bar{e}}_{2}

e_{3} \otimes {\bar{e}}_{3} - e_{2} \otimes {\bar{e}}_{2}, e_{2} \otimes {\bar{e}}_{2} - e_{1} \otimes {\bar{e}}_{1} .

Finally we compute the weights of these vectors: the weights of

e_{1}, e_{2}, e_{3}

(1, 0), (- 1, 1), (0, - 1)

and the weights of

{\bar{e}}_{1}, {\bar{e}}_{2}, {\bar{e}}_{3}

(- 1, 0), (1, - 1), (0, 1)

. Since the weights of

e_{j} \otimes {\bar{e}}_{k}

is the sum of the weights of

e_{j}

{\bar{e}}_{k}

, we get the weights:

\begin{array}{rcl} (1, 0) + (1, - 1) = (2, - 1), (1, 0) + (0, 1) = (1, 1), (- 1, 1) + (- 1, 0) = (- 2, 1), \\ (- 1, 1) + (0, 1) = (- 1, 2), (0, - 1) + (- 1, 0) = (- 1, - 1), (0, 1) + (1, - 1) = (1, - 2), \end{array}

and the last two vectors have the weight

(0, 0)

. Here comes a diagrammatic visualization:

π (Y_{1})

π (Y_{2})

π (X_{1})

π (X_{2})

are again depicted as blue, cyan, red and green arrows. But this time the weight space with weight

(0, 0)

is two dimensional, thus it has two basis vectors and the numbers attached to an outgoing arrow from this space denote the multiples of the weight vector, these vectors are mapped on. Also an arrow pointing to the weight space with weight

(0, 0)

has an additional number put in parentheses, indicating the first or the second basis vector of the weight space.

8-representation

In particle physics the vectors

e_{1} \otimes {\bar{e}}_{1}, \dots, e_{3} \otimes {\bar{e}}_{3}

are designated as

u \bar{u}, \dots, s \bar{s}

, which again are particles composed of a quark and an antiquark and are usually called mesons . This notation saves you a lot of writting!

(s \bar{s} - d \bar{d}) / \sqrt{2}

(d \bar{d} - 2 u \bar{u} + s \bar{s}) / \sqrt{6}

is an orthonormal basis of the weight space with weight

(0, 0)

π (Y_{1})

π (Y_{2})

π (X_{1})

π (X_{2})

with respect to this basis.

Compute the charge of the states

u \bar{d}, u \bar{s}, d \bar{u}, d \bar{s}, s \bar{u}, s \bar{d}

(u \bar{u} - d \bar{d}) / \sqrt{2}

(d \bar{d} - u \bar{u}) / \sqrt{2}

We have to compute

π (Q) u \bar{d} = (Q u) \bar{d} + u (\bar{Q} \bar{d})

, which comes down to:

(2 / 3 - 1 / 3) u \bar{d}

. so the charge of

u \bar{d}

1 / 3

Heighest Weight $(3, 0)$ - the $10$ -representation

This time we use the "quark" notation; it will really make the calculations more clearly laid out: The highest weight vector is

u u u

Y_{1}

u

d

d, s

0

Y_{2}

d

s

u, s

0

π (Y_{j}) = Y_{j} \otimes 1 \otimes 1 + 1 \otimes Y_{j} \otimes 1 + 1 \otimes 1 \otimes Y_{j}

j = 1

u u u

recursively gives:

\begin{array}{rcl} u u u & \to & d u u + u d u + u u d, \\ d u u + u d u + u u d & \to & 2 (d d u + d u d + u d d), \\ d d u + d u d + u d d & \to & 3 d d d, \\ d d d & \to & 0 \end{array}

(3, 0), (1, 1), (- 1, 2), (- 3, 3)

π (Y_{2})

recursively gives:

\begin{array}{rcl} u u u & \to & 0 \\ d u u + u d u + u u d & \to & s u u + u s u + u u s, \\ d d u + d u d + u d d & \to & s d u + d s u + s u d + d u s + u s d + u d s, \\ d d d & \to & s d d + d s d + d d s, \\ s u u + u s u + u u s & \to & 0, \\ s d u + d s u + s u d + d u s + u s d + u d s & \to & 2 (s s u + s u s + u s s), \\ s d d + d s d + d d s & \to & 2 (s s d + s d s + d s s), \\ s s u + s u s + u s s & \to & 0, \\ s s d + s d s + d s s & \to & 3 s s s, \\ s s s & \to & 0. \end{array}

π (Y_{1})

to the newly obtained vectors:

\begin{array}{rcl} s u u + u s u + u u s & \to & s d u + s u d + d s u + u s d + d u s + u d s, \\ s d u + d s u + s u d + d u s + u s d + u d s & \to & 2 (s d d + d s d + d d s), \\ s d d + d s d + d d s & \to & 0, \\ s s u + s u s + u s s & \to & s s d + s d s + d s s, \\ s s d + s d s + d s s & \to & 0, \\ s s s & \to & 0. \end{array}

Since we have not produced any new vector, the irreducible representation lives in the space generated by

10

weight vectors:

\begin{array}{cc} w e i g h t & v e c t o r \\ (3, 0) & u u u \\ (1, 1) & d u u + u d u + u u d \\ (- 1, 2) & d d u + d u d + u d d \\ (- 3, 3) & d d d \\ (2, - 1) & s u u + u s u + u u s \\ (0, 0) & s d u + d s u + s u d + d u s + u s d + u d s \\ (1, - 2) & s s u + s u s + u s s \\ (- 2, 1) & s d d + d s d + d d s \\ (- 1, 1) & s s d + s d s + d s s \\ (1, - 1) & s s s \end{array}

Compute the isospin, the hypercharge and the charge of the states

(d u u + u d u + u u d) / \sqrt{3}

(d d u + d u d + u d d) / \sqrt{3}

Particles made of three quarks are usually called baryons .

Irreducible representation of $S U (3)$ in $P_{n}$

P_{n}

be the homogeneous polynomials on

C^{3}

n \in N_{0}

Γ : S U (3) \to H o m (P_{n})

the representation

Γ (U) f : = f \circ U^{- 1}

. Let us first compute

γ (X) : = D Γ (1) X

: For any holomorphic function

f : C^{3} \to C

we have (cf. section)

γ (X) f (z) = {\frac{d}{d t} |}_{t = 0} f (e^{- t X} z) = - (\partial_{1} f (z), \partial_{2} f (z), \partial_{3} f (z)) X z

and thus

\begin{array}{ccc} γ (X_{1}) f (z) = - \partial_{1} f (z) z_{2}, & γ (X_{2}) f (z) = - \partial_{2} f (z) z_{3}, & γ (X_{3}) f (z) = - \partial_{1} f (z) z_{3}, \\ γ (Y_{1}) f (z) = - \partial_{2} f (z) z_{1}, & γ (Y_{2}) f (z) = - \partial_{3} f (z) z_{2}, & γ (Y_{3}) f (z) = - \partial_{3} f (z) z_{1} . \end{array}

Next we will verify that any non trivial invariant subspace contains the polynomial

z_{3}^{n}

:

γ (X_{2})

and

γ (X_{3})

act on a polynomial by decreasing the degree in

z_{2}

and

z_{1}

, respectively, by one and increasing the degree of

z_{3}

by one.

z_{1}^{n_{1}} z_{2}^{n_{2}} z_{3}^{n_{3}}

will be mapped by

γ (X_{3})^{n_{1}}

to a non zero multiple of

z_{2}^{n_{2}} z_{3}^{n_{1} + n_{3}}

and this gets mapped by

ψ (X_{2})^{n_{2}}

to a non zero multiple of

z_{3}^{n}

. All monomials

z_{1}^{l_{1}} z_{2}^{l_{2}} z_{3}^{l_{3}}

with

l_{1} < n_{1}

or

l_{2} < n_{2}

are mapped to

0

. Now if

f

is a polynomial in an invariant subspace it must be a linear combination of monomials; we pick the one with the lowest degree

n_{3}

, say. Then

ψ (X_{2})^{n_{2}} γ (X_{3})^{n_{1}} p

is just a non zero multiple of

z_{3}^{n}

. The space generated by

ψ (Y_{2})^{l_{2}} γ (Y_{3})^{l_{1}} z_{3}^{n}

is just

P_{n}

, i.e.

γ

and thus

Γ

is irreducible. Eventually

γ (X_{1}) z_{3}^{n} = γ (X_{2}) z_{3}^{n} = 0

, i.e.

z_{3}^{n}

is a highest weight cyclic vector with weight

(0, n)

, because

γ (H_{1}) z_{3}^{n} = 0

,

γ (H_{2}) z_{3}^{n} = - (z_{2} \partial_{2} z_{3}^{n} - z_{3} \partial_{3} z_{3}^{n}) = n z_{3}^{n}

.

The Weyl Group

Once a weight vector of a representation $ψ$ is known, we have to find other weights by applying the operations $ψ (Y_{j})$ or $ψ (X_{j})$ ; but at the time being we don't know if this gives us another weight vector or null. The Weyl group is some sort of a symmetry group of the weights and thus given any weight - we don't need to know a corresponding weight vector - the Weyl group gives us other weights, which, by the way, have also the same multiplicity. We start by another more geometric approach to weights: The subalgebra $H$ generated by $H_{1}$ and $H_{2}$ is a maximal commutative subalgebra of $s l (3, C)$ , it's called the Cartan subalgebra of $s l (3, C)$ . The set $N$ of all $g \in S U (3)$ that leave $H$ invariant under the adjoint representation $A d$ , i.e. $A d (g) (H) \subseteq H$ - this is called the normalizer of $H$ and it's obviously a subgroup of $S U (3)$ (cf. section). Let us investigate this subgroup more closely: for any $g \in N$ the matrices $g H_{j} g^{- 1}$ are in $H$ , hence they must be diagonal, i.e. the standard basis vectors $e_{1}, e_{2}, e_{3}$ are eigen-vectors. Since on the other hand the eigen-vectors of $g H_{j} g^{- 1}$ are $g e_{1}, g e_{2}$ and $g e_{3}$ , we infer that $g e_{1}, g e_{2}$ and $g e_{3}$ must be a permutation of the standard basis $e_{1}, e_{2}$ and $e_{3}$ up to factors of modulus $1$ , i.e.: there are $θ_{1}, θ_{2}, θ_{3} \in R$ and a permutation $π \in S_{3}$ such that $g e_{j} = e^{i θ_{j}} e_{π (j)} and s i g n (π) e^{i \sum θ_{j}} = 1 .$ Conversely, any such $g$ lies in $N$ . A weight $λ$ was defined as a pair of complex numbers: the eigen-values of $ψ (H_{1})$ and $ψ (H_{2})$ for some common eigen-vector $x$ , but in fact for every $H \in H$ we have $H = a_{1} H_{1} + a_{2} H_{2}$ and $λ (H) : = a_{1} λ (H_{1}) + a_{2} λ (H_{2})$ is the eigen-value of $H$ with eigen-vector $x$ . Thus we may conveniently think of a weight as a linear functional $λ$ on the space $H$ . Moreover, endowing $H$ with the euclidean product: $⟨ H, G ⟩ : = t r (H G^{*})$ we know that any linear functional $λ$ can be written as $λ (H) = ⟨ H, G ⟩$ for a unique $G \in H$ . Thus we've got our new definition: a weight of a representation $ψ$ of $s l (3, C)$ in some finite dimensional vector-space $E$ is a vector $λ \in H$ such that there exists some $x \in E ∖ {0}$ such that $\forall H \in H : ψ (H) x = ⟨ H, λ ⟩ x$ Hence weights are particular vectors in the Cartan algebra $H$ of $s l (3, C)$ !

λ_{j} \in H

j = 1, 2

⟨ H_{j}, λ_{k} ⟩ = δ_{j k}

λ_{1}, λ_{2}

is the dual basis to

H_{1}, H_{2}

. Beware, non of them is an orthogonal basis! 2. Show that

H_{3} : = - H_{1} - H_{2}

is orthogonal to

λ_{3} : = λ_{1} - λ_{2}

λ_{1} = d i a g {x, y, z}

x + y + z = 0

x - y = 1

y - z = 0

y = - 1 / 3

z = - 1 / 3

x = 2 / 3

λ_{1}

is the electric charge operator

Q

; analogously we get:

λ_{2} = d i a g {1 / 3, 1 / 3, - 2 / 3}

which is the hypercharge operator

Y

⟨ H_{1} + H_{2}, λ_{1} + λ_{2} ⟩ = 1 - 0 + 0 - 1 = 0

g \in N

H, G \in H

⟨ A d (g) H, A d (g) G ⟩ = t r (g H g^{- 1} (g G g^{- 1})^{*}) = t r (g H G^{*} g^{- 1}) = t r (H G^{*}) = ⟨ H, G ⟩

A d (g)

g \in N

, acts isometrically on

(H, ⟨ ., . ⟩)

. Also, roots are weights of the adjoint representation; hence, in our new interpretation, a root is an element $r \in H$ such that for some root vector $X \in s l (3, C)$ :

\forall H \in H : a d (H) X = ⟨ r, H ⟩ X .

r_{1}, r_{2} \in H

⟨ H_{1}, r_{1} ⟩ = 2

⟨ H_{2}, r_{1} ⟩ = - 1

⟨ H_{1}, r_{2} ⟩ = - 1

⟨ H_{2}, r_{2} ⟩ = 2

λ_{1}, λ_{2}

is the dual basis to

H_{1}, H_{2}

, we must have:

r_{1} = 2 λ_{1} - λ_{2} = d i a g {1, - 1, 0} = H_{1}

r_{2} = - λ_{1} + 2 λ_{2} = d i a g {0, 1, - 1} = H_{2}

Prove that for all

j = 1, 2

H \in H

a d (H) X_{j} = ⟨ H, H_{j} ⟩ X_{j}

a d (H) Y_{j} = ⟨ H, - H_{j} ⟩ X_{j}

H_{j}

- H_{j}

, respectively, are roots and the corresponding root vectors are

X_{j}

Y_{j}

, respectively.

X_{1}

(2, - 1)

, which is in our new interpretation:

2 λ_{1} - λ_{2} = H_{1}

; similarly the root of

X_{2}

(- 1, 2)

, which now becomes

- λ_{1} + 2 λ_{2} = H_{2}

Y_{1}

(- 2, 1)

, which now becomes

- 2 λ_{1} + λ_{2} = - H_{1}

and finally the root of

Y_{2}

(1, - 2)

, which now becomes

λ_{1} - 2 λ_{2} = - H_{2}

H_{1}, H_{2}

H_{3} : = - H_{1} - H_{2}

form an equilateral triangle in

H

λ \in H

is the highest weight, then all other weights

μ

must be of the form

λ - n_{1} H_{1} - n_{2} H_{2}

n_{j} = ⟨ λ - μ, λ_{j} ⟩ \in N_{0}

Ψ : S U (3) \to G l (E)

be any finite dimensional representation with derivative

ψ : = D Ψ (1)

λ \in H

is a weight for

ψ

g \in N

A d (g) λ

is also a weight for

ψ

with the same multiplicity.

Proof :

First we show, that for all

g \in N

Ψ (g) x

is a weight vector with weight

A d (g) λ

H \in H

g^{- 1} H g \in H

ψ (g^{- 1} H g) = Ψ (g)^{- 1} ψ (H) Ψ (g)

(cf. example) and thus

\begin{array}{rcl} ψ (H) Ψ (g) x & = & Ψ (g) Ψ (g)^{- 1} ψ (H) Ψ (g) x \\ = & Ψ (g) ψ (g^{- 1} H g) x = Ψ (g) (⟨ g^{- 1} H g, λ ⟩ x) = ⟨ g^{- 1} H g, λ ⟩ Ψ (g) x \\ = & ⟨ A d (g^{- 1}) H, λ ⟩ Ψ (g) x = ⟨ H, A d (g) λ ⟩ Ψ (g) x, \end{array}

where the last equality follows from the fact that

A d

acts isometrically on

(H, ⟨ ., . ⟩)

. Therefore

Ψ (g)

maps the weight space with weight

λ

into the weight space with weight

A d (g) λ

and since

A d

and

Ψ

are representation

Ψ (g^{- 1})

is the inverse of this map, i.e. the dimensions of the weight spaces with weight

λ

and

A d (g) λ

coincide.

qed

Of course only those elements

g \in N

are of interest for which

A d (g) | H

is different from the identity, thus we factor the kernel

Z

of

A d : N \to G l (H)

:

Z

is given by

Z = {g \in N : \forall H \in H : A d (g) H = H},

which is the centralizer of

H

in

N

, cf. section.

Z

is the set of all diagonal matrices

g \in N

satisfying

g e_{j} = e^{i θ_{j}} e_{j} and \sum θ_{j} = 0 .

Factorizing

A d : N \to G l (H)

gives us an injective homomorphism

N / Z \to G l (H)

. The group

N / Z

is called the Weyl group of

S U (3)

; it's isomorphic to

S_{3}

: to every permutation

π \in S_{3}

we associate it's permutation matrix and, in order to get a determinant one matrix, we multiply by

s i g n (π)

. We define the action of

N / Z

on

H

by

w \cdot H : = A d (g) H

, where

[g] = w

is identified with a permutation of three elements. Let us compute the action of the permutation

w = (231)

on

H

:

A d (g) H = (\begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}) (\begin{array}{ccc} h_{1} & 0 & 0 \\ 0 & h_{2} & 0 \\ 0 & 0 & h_{3} \end{array}) (\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}) = (\begin{array}{ccc} h_{3} & 0 & 0 \\ 0 & h_{1} & 0 \\ 0 & 0 & h_{2} \end{array}) .

In general (cf. example)

w \cdot H

permutes the entries of

H

:

h_{j} \to h_{w^{- 1} (j)}

.

Ψ : S U (3) \to G l (E)

be any finite dimensional representation with derivative

ψ : = D Ψ (1)

λ = d i a g {h_{1}, h_{2}, h_{3}}

is a weight for

ψ

, then for any permutation

w \in S_{3}

d i a g {h_{w (1)}, h_{w (2)}, h_{w (3)}}

is also a weight for

ψ

with the same multiplicity.

The permutations

w \in {(213), (132), (321), (231), (312)}

act on the pair of vectors

(H_{1}, H_{2})

, yielding pairs

(w \cdot H_{1}, w \cdot H_{2})

, which are given by

(- H_{1}, - H_{3}), (- H_{3}, - H_{2}), (- H_{2}, - H_{1}), (H_{3}, H_{1}), (H_{2}, H_{3}) .

The action of the first permutation

(213)

is a reflection about the line

R λ_{1} = [⟨ ., H_{2} ⟩ = 0]

, the action of the second is a reflection about the line

R λ_{2} = [⟨ ., H_{1} ⟩ = 0]

and the action of the third is a reflection about the line

R (λ_{1} - λ_{2}) = [⟨ ., H_{3} ⟩ = 0]

. The forth action is a rotation by

4 π / 3

and the fifth is a rotation by

2 π / 3

. Hence the Weyl group is just the symmetry group of the regular triangle formed by

λ_{3} : = λ_{1} - λ_{2}, λ_{2}, - λ_{1}

and the reflections about the three lines orthogonal to

H_{j}

j = 1, 2, 3

generate this group of symmetries. That's the way we look at the Weyl group: a subgroup of the group of isometries of the two dimensional space

H

. The picture below illustrates the lattice points

Z λ_{1} + Z λ_{2}

(which contains all possible weights), the regular triangle (of which the Weyl group is the symmetry group) and the positive sector

P : = R_{0}^{+} λ_{1} + R_{0}^{+} λ_{2} = [⟨ ., H_{1} ⟩ \geq 0] \cap [⟨ ., H_{2} ⟩ \geq 0]

of non negative weights bounded by half-lines through

λ_{1}

λ_{2}

. In addition, the sector indicates all weights lower than

- H_{3}

weyl1

Some conclusions can be drawn from this interpretation immediately: 1. If

λ \neq 0

is any weight, then the set

{w \cdot λ : w \in N / Z}

contains either three or six points, depending on whether

λ

lies on one of the lines generated by

λ_{1}, λ_{2}

λ_{1} - λ_{2}

or not.
2. The vectors

\pm H_{j}

j = 1, 2, 3

, form a regular hexagon and for every lattice point in

λ \in Z λ_{1} + Z λ_{2}

w

N / Z

w \cdot λ

lies in the positive sector

P

λ_{1}

λ_{2}

⟨ w \cdot λ, H_{1} ⟩

⟨ w \cdot λ, H_{2} ⟩

are non negative.

1

-representation: Suppose we have a representation

ψ = D Ψ (1)

, which has the weight

(1, 0)

, which now becomes

λ_{1} = d i a g {2, - 1, - 1} / 3

, then it must also have the weights

d i a g {- 1, 2, - 1} / 3

d i a g {- 1, - 1, 2} / 3

, which in old notation is

(- 1, 1)

(0, - 1)

8

-representation: This representation has the weight

(1, 1)

λ_{1} + λ_{2} = - H_{3} = d i a g {1, 0, - 1}

, then it must have another five weights:

H_{3}, \pm H_{1}, \pm H_{2}

, which can be written as

- λ_{1} + 2 λ_{2}, - λ_{1} - λ_{2}, 2 λ_{1} - λ_{2}, - 2 λ_{1} + λ_{2}, λ_{1} - 2 λ_{2},

or in old notation:

(- 1, 2), (- 1, - 1), (2, - 1), (- 2, 1)

(1, - 2)

.
Assume a representation

ψ = D Ψ (1)

(0, 3)

(1, 1)

. What additional weights

ψ

must have? These weights in our new approach are given by

3 λ_{1}

- H_{3}

, thus it must have the additional weights:

3 (λ_{1} - λ_{2})

- 3 λ_{1}

H_{3}, \pm H_{1}, \pm H_{2}

Assume a representation

ψ = D Ψ (1)

(3, 2)

(1, 2)

. What are the additional weights

ψ

λ

be the highest weight of the irreducible representation

ψ

w_{3}

be the reflection about the line

R λ_{3} = [⟨ ., H_{3} ⟩ = 0]

w_{3} \cdot λ

is the lowest weight of

ψ

- w_{3} \cdot λ

is the highest weight of its dual

\bar{ψ}

Decompose the adjoint representation

a d : s l (3, C) \to H o m (s l (3, C))

into irreducible representations.

The Weyl group as a subgroup of the isometries on the Cartan algebra

H

does not contain the inversion. Show that

ψ

\bar{ψ}

are equivalent iff the weights of

ψ

are invariant under the inversion.

Suppose an irreducible representations has highest weight

λ

with weight vector

x_{0}

. Prove that the weight space with weight

λ - H_{1} - H_{2}

x_{1} : = ψ (Y_{1}) ψ (Y_{2}) x

x_{2} : = ψ (Y_{2}) ψ (Y_{1}) x

and is thus at most two dimensional.

E

carries an euclidean product

⟨ ., . ⟩

such that we have for an irreducible representations

ψ : s l (3, C) \to H o m (E)

π (X)^{*} = - π (X)

X \in s u (3)

λ = (m_{1}, m_{2})

be the highest weight of

ψ

with weight vector

x_{0}

ψ (X_{j})^{*} = ψ (Y_{j})

⟨ u_{1}, u_{1} = m_{2} (m_{1} + 1)

⟨ u_{2}, u_{2} = m_{1} (m_{2} + 1)

⟨ u_{1}, u_{2} ⟩ = m_{1} m_{2}

m_{1} \geq 1

m_{2} \geq 1

x_{1}

x_{2}

are linearly independent. 4. If

m_{1} = 0

m_{2} \geq 1

or conversely, then

x_{1}

x_{2}

are linearly dependent.

X_{j} - Y_{j}, i X_{j} + i Y_{j} \in s u (3)

\begin{array}{rcl} π (X_{j})^{*} - ψ (Y_{j})^{*} & = & - π (X_{j}) + π (Y_{j}) and \\ - i π (X_{j})^{*} - i π (Y_{j})^{*} & = & - i π (X_{j}) - i π (Y_{j}) \end{array}

π (X_{j})^{*} = π (Y_{j})

Convex hulls of weights of irreduvible representations

λ

is the highest weight of an irreducible representation

ψ

. Then the convex hull

C

{w \cdot λ : w \in N / Z}

contains all weights of

ψ

. Indeed, we have just seen that there is some

w \in N / Z

such that for any weight

μ

ψ

w \cdot μ \in P

λ

about the lines

R λ_{1} = [⟨ ., H_{2} ⟩ = 0]

R λ_{2} = [⟨ ., H_{1} ⟩ = 0]

gives us two points

w_{1} \cdot λ

w_{2} \cdot λ

on the boundary of

[⟨ ., λ_{1} ⟩ \leq ⟨ λ, λ_{1} ⟩] \cap [⟨ ., λ_{2} ⟩ \leq ⟨ λ, λ_{2} ⟩] .

convex

w \cdot μ

is in the intersection

D

P

with the convex hull of

0, w_{1} \cdot λ, w_{2} \cdot λ, λ

D = P \cap [⟨ ., λ_{1} ⟩ \leq ⟨ λ, λ_{1} ⟩] \cap [⟨ ., λ_{2} ⟩ \leq ⟨ λ, λ_{2} ⟩],

which follows from the fact that

w \cdot μ

λ

⟨ w \cdot μ, λ_{j} ⟩ \leq ⟨ λ, λ_{j} ⟩

0 = \frac{1}{6} \sum_{w} w \cdot λ \in C

D \subseteq C

μ \in ⋃ w \cdot D \subseteq ⋃ w \cdot C = C

μ

be a weight of an irreducible representation

ψ : s l (3, C) \to H o m (E)

w \in N / Z

is a reflection about the line orthogonal to

H_{j}

j = 1, 2, 3

, then any point on the line segment joining

μ

w \cdot μ

μ + Z H_{j}

ψ

Proof :

The subalgebras

A_{j} : = s p a n (H_{j}, X_{j}, Y_{j})

j = 1, 2, 3

are all isomorphic to

s l (2, C)

. W.l.o.g we may assume

j = 1

F

be the subspace of

E

generated by all weight vectors in

E

whose weights lie on the line

μ + R H_{1}

. These weights are shifted by

ψ (X_{1})

ψ (Y_{1})

\pm H_{1}

and therefore the restrictions

ψ | A_{1}

gives a representation

φ

s l (2, C)

F

w \cdot H_{1} = - H_{1}

w^{*} = w

⟨ H_{1}, w \cdot μ ⟩ = ⟨ w \cdot H_{1}, μ ⟩ = - ⟨ H_{1}, μ ⟩ .

φ (H_{1})

has the eigen-values

- | ⟨ H_{1}, μ ⟩ |, - | ⟨ H_{1}, μ ⟩ | + 2, \dots, | ⟨ H_{1}, μ ⟩ | - 2, | ⟨ H_{1}, μ ⟩ |,

which coincides with the set

{⟨ H_{1}, ν ⟩}

, where

ν

is the set of points

ν = μ + Z H_{1}

on the line segment joining

μ

to

w \cdot μ

. Thus for any such

ν

there must be an eigen-vector

x \in F

of

H_{1}

such that:

H_{1} x = ⟨ H_{1}, ν ⟩ x

.

qed

ψ : s l (3, C) \to H o m (E)

be an irreducible representation with highest weight

λ

μ

ψ

if and only if the following three conditions hold:

$μ \in Z λ_{1} + Z λ_{2}$ ,
$μ \in λ - N_{0} H_{1} - N_{0} H_{2}$ ,
$μ \in C : = c o (w \cdot λ : w \in N / Z)$ ,

Proof :

We only need to prove that these conditions are sufficient. By lemma each point in

λ - N_{0} H_{1} - N_{0} H_{2}

on the boundary of

C

is a weight. So assume

μ = λ - n_{1} H_{1} - n_{2} H_{2}

is in the interior of

C

; w.l.o.g. we may also assume:

n_{2} \leq n_{1}

, then

μ = λ - m H_{1} + n_{2} H_{3}, with m = n_{1} - n_{2} .

Starting at

μ

in the direction of

- H_{3}

we end up in

ν : = λ - (n_{1} - n_{2}) H_{1}

after

n_{2}

steps.

ν

lies on the boundary of the convex hull

C

of

{w \cdot λ : w \in N / Z}

, for it cannot lie on a boundary part of

C

parallel to

H_{3}

an the intersection with the line passing through

λ

and parallel to

H_{2}

is given by the equation

λ - m_{1} H_{1} + x_{3} H_{3} = λ - x_{2} H_{2}

; putting

a_{j} = x_{j} / m_{1}

gives

H_{1} = a H_{2} + b H_{3}

, i.e.

a = b = - 1

and thus

λ - x_{2} H_{2} = λ + m_{1} H_{2}

, which is not in

C

. Thus

ν

is a weight and so is its reflection

w \cdot ν

about the line orthogonal to

H_{3}

; by lemma

μ

must also be a weight.

qed

Assume a irreducible representation

ψ : s l (3, C)

has highest weight

4 λ_{1} + 2 λ_{2}

. Identify all additional weights of

ψ

← Matrix Lie-Groups → Topological Groups

Last modified: Mon Oct 23 16:04:06 CEST 2023