9. Casimir operators, characters, dimension and strange formulae

* version 1.6 *

Matthew Foster

January 3, 2019

Contents

9.1 Quadratic Casimir operator 19.1.1 Standard normalization for Killing form and Casimir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.1.2 Index of a representation; some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

9.1.2.1 su(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79.1.2.2 su(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

9.2 Freudenthal’s recursion formula for weight space dimensions 8

9.3 Weyl group 11

9.4 Weyl’s character formula: derivation 139.4.1 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139.4.2 Freudenthal redux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.4.3 Resumming Freudenthal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.4.4 finis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9.5 Dimension and strange formulae 209.5.1 Dimensions for generic representations of the classical algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

The discussion here follows chapters XI–XIII of [1] and Secs. 13.2 and 13.4 of [2].

9.1 Quadratic Casimir operator

An essential datum for each irreducible su(2) representation is the eigenvalue C(2j) = j(j + 1) of the quadratic Casimir operator(see Sec. 1.3.3),

Ĉ ≡ T̂zT̂z +1

2

(T̂+T̂− + T̂−T̂+

), (9.1.1)

where 2j ∈ {0, 1, 2, . . .} is the Dynkin label for the spin-j representation (i.e., the Dynkin coefficient of the highest weight). The keyproperties of the Casimir operator are

1. It gives the same eigenvalue for all states in an irreducible representation,

Ĉ |j, m〉 = j(j + 1) |j, m〉 , T̂z |j, m〉 = m |j, m〉 , m ∈ {−j,−j + 1, . . . , j − 1, j}. (9.1.2a)

2. It commutes with all generators,

[Ĉ, T̂+] = [Ĉ, T̂−] = [Ĉ, T̂z] = 0. (9.1.2b)

1

Eigenvalues of an su(2)-invariant Hamiltonian can depend only on C(2j) = j(j + 1), not on the “magnetic quantum number” m thatdistinguishes different weight vectors. Although each representation is uniquely labeled by its Casimir eigenvalue, it is important toremember that Ĉ is not defined abstractly for the Lie algebra A1 = su(2), but only for a particular representation in terms of theexplicit bilinear sum of matrix generators in Eq. (9.1.1).1

For a generic rank-n Lie algebra, it turns out that one can construct exactly n Casimir operators satisfying the axioms inEq. (9.1.2) [3]. We will discuss only the generalization of the quadratic Casimir in Eq. (9.1.1), which is unique. The other higherorder Casimir operators are built from sums of m-fold products, where m ≥ 3 is the order of each factor in the operator.2

We define the quadratic Casimir operator for a rank-n Lie algebra L as an operator acting on states of a generic finite-dimensional, irreducible representation as follows:

Ĉ ≡n∑

i,j=1

A−1ij Ĥαi Ĥαj +∑

α 6=0

Êα Ê−α(eα, e−α)

, Quadratic Casimir operator,

Aij ≡ 〈αi, αj〉 = Aij|αj|2

2.

(9.1.3a)

(9.1.3b)

Just as in su(2) [Eq. (9.1.1)], the operator Ĉ is a sum of bilinears of Cartan subalgebra elements {Ĥαi} or root vector raising andlowering operators {Êα, Ê−α}. The overlap matrix Aij is a symmetrized version of the Cartan matrix Aij = 〈αi, α∨j 〉 [Eq. (5.3.8)].Note that the sum over root vector operators in Ĉ includes all nonzero roots, not just positive or simple ones. The number of termsin the sum Eq. (9.1.3a) is equal to Nθ, the dimension of the adjoint representation: [recall that θ labels the highest weight (root) ofthe adjoint representation, Secs. 5.3.1 and 8.2.4]

Nθ = n + |∆| = n + |∆+|+ |∆−| = d. (9.1.4)

In this equation, n denotes the rank, |∆| (|∆±| = |∆|/2) is the number of roots (number of positive or negative roots), and d is thedimension of the corresponding continuous group G generated by the Lie algebra L.Claim I: The quadratic Ĉ is a pure number in any irreducible representation. I.e., it commutes with all {Ĥαi , Êα}.Before we prove this statement, we examine some of its consequences.

1. Trace in the adjoint representation. Consider the trace

(C) ≡ Tr[Ĉ

]

θ, (9.1.5)

where Tr[· · · ]Λ denotes the trace in representation Λ; here θ indicates the adjoint representation. Since the trace in the adjointrepresentation is the Killing form Tr[âdx âdy]θ = (x, y), we have

(C) =

n∑

i,j=1

A−1ij(hαi , hαj

)+

∑

α 6=0

=

n∑

i,j=1

A−1ij Aji +∑

α 6=0

= Nθ . (9.1.6)

2. Action on the highest-weight state φΛ of an irreducible representation. Let Λ denote the associated highest weight. Then

ĈφΛ =

n∑

i,j=1

〈Λ, αi〉A−1ij 〈αj , Λ〉φΛ +∑

α>0

[Ê−α Êα(eα, e−α)

+ Ĥα

]φΛ

=

[〈Λ, Λ〉 +

∑

α>0

〈Λ, α〉]

φΛ. (9.1.7)

On the first line of Eq. (9.1.7), we have restricted the sum over root vectors to positive roots (because φΛ is a highest weightstate annihilated by all Ê−α with α < 0), and we have employed the definition for hα in Eq. (4.4.10). On the second line, wehave removed the resolution of the identity in the rank-n space of weights H∗. For a linearly independent, but not necessarilyorthonormal basis of states {|i〉}, the identity can be expressed as

1̂n =n∑

i,j=1

O−1ij |i〉 〈j| , Oij = 〈i |j〉 . (9.1.8)

1To write Eq. (9.1.1), we need to define the (matrix) product between generators. The only product operation defined for the Lie algebra itself is the Liebracket, which becomes a matrix commutator in a finite-dimensional representation.

2The order m + 1 of the (m + 1)-fold products summed in a particular Casimir is called the exponent m. For each algebra An, Bn, Cn, and Dn, the nexponents of the algebra are tabulated (e.g.) in Sec. 13A of [2].

2

It will prove useful to rewrite Eq. (9.1.7) in a more compact way. We define the Weyl vector for a Lie algebra as the weight

ρW ≡1

2

∑

α>0

α =

n∑

j=1

ωj, Weyl vector for a rank-n Lie algebra. (9.1.9)

The second equality implies that the Dynkin coefficients of the Weyl vector are (Λ1, . . . , Λn) = (1, 1, 1, . . . , 1). We will prove this inSec. 9.3, below. Eq. (9.1.7) then becomes

ĈφΛ ≡ CΛ φΛ = 〈Λ, Λ + 2ρW〉φΛ. (9.1.10)

Proof: The quadratic Ĉ commutes with all generators {Ĥαi , Êα} in an irreducible representation. Exploiting the Lie bracket structureof a generic Lie algebra [Eq. (4.4.11)], we have

[Ĉ, Ĥαi

]=

∑

α 6=0

Êα

[Ê−α, Ĥαi

]+

[Êα, Ĥαi

]Ê−α

(eα, e−α)

=∑

α 6=0

Êα Ê−α〈α, αi〉 − 〈α, αi〉Êα Ê−α(eα, e−α)

= 0, (9.1.11a)

[Ĉ, Êβ

]=

n∑

i,j=1

A−1ij 〈β, αj〉(Ĥαi Êβ + Êβ Ĥαi

)+

∑

α 6=0

Êα

[Ê−α, Êβ

]+

[Êα, Êβ

]Ê−α

(eα, e−α)

=

n∑

i,j=1

A−1ij 〈β, αj〉(Ĥαi Êβ + Êβ Ĥαi

)+

∑

α 6=0α 6=β

N−α,β Êα Êβ−α(eα, e−α)

+∑

α 6=0α 6=−β

Nα,β Êα+β Ê−α(eα, e−α)

− Êβ Ĥβ − Ĥβ Êβ. (9.1.11b)

Next we invoke invariance of the Killing form [Eq. (4.4.8)], as well as Eq. (4.4.5) to obtain

(eα, [eβ, eγ ]) =Nβ,γ (eα, e−α) δβ+γ+α,0 = −N−β−α,β (eα, e−α) δβ+γ+α,0=([eα, eβ], eγ) = Nα,β (eα+β , e−α−β) δα+β+γ,0 ,

⇒ Nα−β,β(eβ−α, eα−β)

= − N−α,β(eα, e−α)

. (9.1.12)

In this equation, we have used Nα,β = −Nβ,α, which follows from the antisymmetric property of the Lie bracket [Eq. (1.2.1a)]; onthe last line we have sent α → −α on both sides. We can therefore rewrite the red term in Eq. (9.1.11b) as follows:

∑

α 6=0α 6=−β

Nα,β Êα+β Ê−α(eα, e−α)

=∑

α′ 6=0α′ 6=β

Nα′−β,β Êα′ Êβ−α′

(eα′−β, e−α′+β)= −

∑

α′ 6=0α′ 6=β

N−α′,β Êα′ Êβ−α′

(eα′ , e−α′), (9.1.13)

where we have defined α′ ≡ α + β. The two terms in Eq. (9.1.11b) explicitly involving the structure constants therefore exactlycancel, leaving

[Ĉ, Êβ

]= Êβ

n∑

i,j=1

A−1ij 〈β, αj〉Ĥαi − Ĥβ

+

n∑

i,j=1

A−1ij 〈β, αj〉Ĥαi − Ĥβ

Êβ = 0. (9.1.14)

The final equality follows from the representation of Ĥβ in the linearly independent basis {Hαj}; e.g., consider a state |f〉 multipliedby the resolution of the identity in Eq. (9.1.8):

|f〉 =n∑

i,j=1

O−1ij |i〉 〈j |f〉 .

Q.E.D.

3

9.1.1 Standard normalization for Killing form and Casimir

The Dynkin coefficients for any weight in an irreducible representation are guaranteed to be integers by the MWDF, Eq. (5.1.23).Dynkin coefficients are independent of the normalization of simple roots, simple coroots, or fundamental weights; the same is true ofthe Cartan matrix Aij = 〈αi, α∨j 〉 = 2〈αi, αj〉/〈αj, αj〉, the rows of which give the Dynkin coefficients for the simple roots.

By contrast, the quadratic form matrix Fij = 〈ωi, ωj〉 [Eq. (8.1.5c)] and the quadratic Casimir eigenvalue in Eq. (9.1.10) dodepend upon the normalization scheme. In certain applications (notably affine Lie algebras in conformal field theory (CFT) [2]), itis crucial to adopt a consistent normalization. We define the “standard” normalization scheme used in CFT and elsewhere in thissubsection, and will employ it throughout the remainder of these lectures (unless otherwise noted).

Recall that all roots (including simple roots) come in one or two varieties for semi-simple Lie algebras: simply-laced algebrashave all roots with the same length; otherwise there are “short” and “long” roots (Prop. IV., Sec. 5.2.2). The inner products ofsimple roots were computed for the four classical families in module 7, using a particular normalization scheme for the generators ofthe defining representation in each case. The results for An−1 = su(n) [Eq. (7.1.21)], Bn = so(2n + 1) [Eq. (7.4.19)], Cn = sp(2n)[Eq. (7.2.20)], and Dn = so(2n) [Eq. (7.3.23)] show that each long simple root α

(long)

i for algebras Bn and Cn (all simple roots forsimply-laced algebras, An and Dn) is normalized to

〈α(long)i , α

(long)

i

〉= |α(long)i |2 =

1

g, (9.1.15)

where g is the dual Coxeter number defined in module 8, Eq. (8.1.13). The tabulated values for g in each case appear in Eq. (8.6.1),and match Eqs. (7.1.21), (7.4.19), (7.2.20), and (7.3.23).

The standard normalization scheme resets |α(long)i |2 = 2 for all algebras, i.e. all long roots normalized to two. Thehighest root θ (Sec. 8.2.4) is always a long root, so that this scheme always has |θ|2 = 2. To achieve this, we renormalize the followingobjects:

• Renormalized Killing form:

(x, y)2 ≡1

2g(x, y) =

1

2gTr

[âdx âdy

]θ. (9.1.16a)

The trace in the adjoint representation for a product of generators is 2g times the renormalized Killing form.

• Renormalized root representation in H , the Cartan subalgebra: Let hα ∈ H represent the root α ∈ ∆; ∆ denotes the set ofroots. hα is explicitly defined in the old normalization scheme via Eq. (4.4.10). Then we define the renormalized version

h(2)α ≡ 2g hα ⇒ (h(2)α , k)2 = (hα, k) = α(k). (9.1.16b)

Thus the eigenvalue of a root α acting on a generic element k ∈ H is unchanged.

• Root scalar product: Let α and β belong to the set of roots ∆. Then we have the renormalized root scalar product

〈α, β〉2 ≡(h(2)α , h

(2)

β

)

2= 2g 〈α, β〉. (9.1.16c)

Comparing Eqs. (9.1.15) and (9.1.16c) guarantees that〈α(long)i , α

(long)

i

〉2

= 2, (9.1.17)

as desired. We also define a renormalized Casimir operator [c.f. Eq. (9.1.3a)],

Ĉ2 ≡n∑

i,j=1

A−12 ijĤ (2)αi Ĥ(2)

αj+

∑

α 6=0

Êα Ê−α(eα, e−α)2

, Quadratic Casimir operator (standard normalization),

A2 ij ≡〈αi, αj〉2 = Aij〈αj , αj〉2

2.

(9.1.18a)

(9.1.18b)

It is evident that

Ĉ2 = 2g Ĉ. (9.1.19)

Now, the analog of the trace in Eqs. (9.1.5) and (9.1.6) is

(C2)2 ≡1

2gTr

[Ĉ2

]

θ= (C) = Nθ. (9.1.20)

4

Eqs. (9.1.10) and (9.1.20) then imply that

Cθ2 = 〈θ, θ + 2ρW〉2 = 2g. Standard Casimir in the adjoint representation= twice the dual Coxeter number.

(9.1.21)

If we express θ in terms of its comarks [Eq. (8.1.12)],

θ =

n∑

j=1

a∨j α∨j , (9.1.22)

then using 〈ωi, α∨j 〉 = 〈ωi, α∨j 〉2 = δij, we get

〈θ, ρW〉2 =n∑

j=1

a∨j . (9.1.23)

Here we have used the fact that the Weyl vector ρW always has Dynkin coefficients (1, 1, 1, . . . , 1) [Eq. (9.1.9)]. Combined with thefact that 〈θ, θ〉2 = 2, Eqs. (9.1.21) and (9.1.23) give

g = 1 +

n∑

j=1

a∨j , (9.1.24)

which is the formula for the dual Coxeter number given in terms of comarks by Eq. (8.1.13).We can compute Eq. (9.1.21) explicitly, using (a) the Dynkin coefficients of the highest root θ = (Λ(θ)1 , Λ

(θ)

2 , . . . , Λ(θ)n )

(Sec. 8.2.4), (b) the Dynkin coefficients of the Weyl vector ρW = (1, 1, . . . , 1) [Eq. (9.1.9)], and (c) the quadratic form matrixFij, already tabulated in the standard normalization for each algebra in Eq. (8.6.2). The Casimir for the adjoint representation isthen

Cθ2 = 2 + 2

n∑

i,j=1

Λ(θ)i Fij. (9.1.25a)

For su(n + 1), Λ(θ)1 = Λ(θ)n = 1 and all others vanish [Eq. (8.2.25)], while Fij has the form in Eq. (8.6.2a). This leads to

Cθ (An)2 = 2 + 2

n∑

j=1

(F1j + Fnj) = 2 +2

n + 1[n(n + 1)] = 2(n + 1). (9.1.25b)

For sp(2n), Λ(θ)1 = 2 and all others vanish [Eq. (8.2.27)], while Fij has the form in Eq. (8.6.2c):

Cθ (Cn)2 = 2 + 2

2n∑

j=1

F1j = 2(n + 1). (9.1.25c)

For so(2n), Λ(θ)2 = 1 and all others vanish [Eq. (8.2.26)], while Fij has the form in Eq. (8.6.2d):

Cθ (Dn)2 = 2 + 2

n∑

j=1

F2j = 2 + [3 × 2 + (n − 3)4] = 2(2n − 2). (9.1.25d)

Eqs. (9.1.25b)–(9.1.25d) agree with the previously quoted results for 2g in Eq. (8.6.1).For a generic representation with highest weight Λ = (Λ1, · · · , Λn), the standard Casimir is

CΛ2 = 〈Λ, Λ + 2ρW〉2 = 〈Λ, Λ〉2 + 2n∑

i,j=1

Λi Fi,j, Standard Casimir for highest weight representation Λ. (9.1.26)

5

For the defining representation Λ = ω1, one finds the following

Algebra Casimir 〈ω1, ω1 + 2ρW〉2An = su(n + 1)

n(n+2)n+1

Bn = so(2n + 1) 2n

Cn = sp(2n) n +12

Dn = so(2n) 2n − 1

(9.1.27)

• Exercise: Show that the standard Casimir for a rank-q fully antisymmetric tensor representation of An is given by

〈Λ, Λ + 2ρW〉2 =q(n + 1 − q)(n + 2)

n + 1, 1 ≤ q ≤ n. (9.1.28)

• Exercise: Show that the standard Casimir for a rank-q fully antisymmetric tensor representation of Cn is given by

〈Λ, Λ + 2ρW〉2 =1

2q [2(n + 1) − q] , 1 ≤ q ≤ n. (9.1.29)

9.1.2 Index of a representation; some examples

Let x and y be elements of the Lie algebra, x, y ∈ L. For a highest weight representation Λ, define

((x, y))Λ ≡ Tr[X̂ Ŷ

]

Λ, (9.1.30)

where the right-hand side is the trace normalization of the product of generators in the irreducible representation. The index lΛ forrepresentation Λ is then defined via

((x, y))Λ = lΛ (x, y)2. Index of a representation. (9.1.31)

The index relates the trace normalization in some arbitrary irreducible representation Λ to that in the adjoint representation (i.e., therenormalized Killing form). Using the renormalized Casimir operator constructed in the representation Λ, Eqs. (9.1.18a), (9.1.26),and (9.1.31) imply that

((C2)) = lΛ(C2)2 = lΛ Nθ

= 〈Λ, Λ + 2ρW〉2 NΛ,(9.1.32)

which leads to

lΛ =〈Λ, Λ + 2ρW〉2 NΛ

Nθ. (9.1.33)

Here NΛ is the dimension of the representation Λ (i.e., the total number of states, including degeneracies); Nθ is the dimension ofthe adjoint representation. Note that

lθ = 〈θ, θ + 2ρW〉 = 2g, (9.1.34)

consistent with Eq. (9.1.16a). The index of the defining representation lω1 , the adjoint representation lθ = 2g [Eq. (8.6.1)], as well asthe dimension Nω1 and the dimension of the adjoint Nθ for the classical families are [via Eq. (9.1.27)]

Algebra Nω1 Nθ lω1 lθ = 2g

An = su(n + 1) n + 1 n2 + 2n 1 2(n + 1)

Bn = so(2n + 1) 2n + 1 n(2n + 1) 2 2(2n − 1)Cn = sp(2n) 2n (2n)(n + 1/2) 1 2(n + 1)

Dn = so(2n) 2n n(2n − 1) 2 4(n − 1)

(9.1.35)

For the orthogonal algebras, it is interesting to compute the Casimirs and indices for the spinor representations.• Exercise: Show that the standard Casimir for the ωn fundamental spinor representation of Bn = so(2n + 1) is given by

〈ωn, ωn + 2ρW〉2 =n

4(2n + 1). (9.1.36)

6

• Exercise: Show that the standard Casimir for the ωn−1 and ωn fundamental spinor representations of Dn = so(2n) is given by

〈ωn−1, ωn−1 + 2ρW〉2 = 〈ωn, ωn + 2ρW〉2 =n

4(2n − 1). (9.1.37)

Using the facts (proven later) that Nωn = 2n in Bn and Nωn−1 = Nωn = 2

n−1 for Dn, the indices are given by

Algebra Fundamental spinor rep. index(ices)

Bn = so(2n + 1) lωn = 2n−2

Dn = so(2n) lωn−1 = lωn = 2n−3

(9.1.38)

9.1.2.1 su(2)

In su(2), the Cartan matrix is a single number: A = 〈θ, θ〉 = 2. For a simply-laced algebra, α∨ = 2α/|α|2 = α in the standardnormalization scheme. Then the quadratic form matrix is F = 〈ω1, ω1〉 = A−1 = 1/2. The Weyl vector is ρW = ω1. The spin jrepresentation has highest weight Λ = 2j. The Casimir for this representation is

C(2j)2 = 2j(2j + 2)〈ω1, ω1〉2 = 2j(j + 1). (9.1.39)

This is twice the usual normalization in quantum mechanics. The index is

l2j = 2j(j + 1)2j + 1

3=

2j(j + 1)(2j + 1)

3. (9.1.40)

The index of the first few j values is

j l2j0 01/2 11 43/2 102 205/2 35

(9.1.41)

Eq. (9.1.41) implies that the su(2) generators in the defining j = 1/2 representation are

T̂x,y,z =1√2σ̂x,y,z,

[T̂i, T̂j

]= if ijkT̂k. (9.1.42)

where σ̂x,y,z are the standard Pauli matrices. The structure constants are f ijk =√

2�ijk, which explains the “extra” factor of two inEq. (9.1.39).

9.1.2.2 su(3)

A special feature of su(3) is that any highest weight representation can be associated to a traceless bisymmetrized tensor T̃(j1···jq)

(i1···ip),

T̃(kj2···jq)(ki2···ip)

= 0. The Dynkin coefficients of the associated highest weight are (p, q) [c.f. Eq. (8.5.2)]. The dimension is the number of

independent elements of this tensor,

Np,q =(p + 2)!

p!2!

(q + 2)!

q!2!− (p + 1)!

(p − 1)!2!(q + 1)!

(q − 1)!2! =(p + 1)(q + 1)(p + q + 2)

2. (9.1.43)

This result can also be obtained from the “factors over hooks” rule. The Casimir for this representation is

C(p,q)2 = 〈p ω1 + q ω2, (p + 2)ω1 + (q + 2)ω2〉

=2

3[p(p + 2) + q(q + 2)] +

1

3[p(q + 2) + q(p + 2)] =

2

3

[p2 + q2 + pq + 3(p + q)

]. (9.1.44)

Thus the index is

lp,q =C(p,q)2 Np,q

Nθ=

1

24(p + 1)(q + 1)(p + q + 2)

[p2 + q2 + pq + 3(p + q)

]. (9.1.45)

7

The Casimir and index for a few low-dimensional representations are

HWSconjugacy

classNp,q C

(p,q)

2 lp,q

(0, 0) 0 1 0 0(1, 0) 1 3 8/3 1(1, 1) 0 8 6 6(2, 0) 2 6 20/3 5(2, 1) 1 15 32/3 20(2, 2) 0 27 16 54

(9.1.46)

Eqs. (9.1.35), (9.1.38), (9.1.41), and (9.1.46) suggest that the index is always a non-negative integer; this is indeed the case.

9.2 Freudenthal’s recursion formula for weight space dimensions

The master weight depth formula Eq. (5.1.23) allows the determination of all weights in a finite-dimensional, highest weight irreduciblerepresentation of a Lie algebra. It does not determine the total number of states in a given representation, however, because someweights can be degenerate. For example, the (2,1) representation of su(3) shown in Fig. 9.1 has only 12 distinct weights. On theother hand, Eq. (9.1.43) implies that N2,1 = 15. Therefore between one to three of the weights in (2,1) must be degenerate. Recallthat the weight space refers to the set of states sharing the same weight, i.e. the same Dynkin coefficients.

In this section we will determine a recursive algorithm (Freudenthal’s formula) to determine the dimension of all weightspaces in a given representation. Let M be a weight in the representation with highest weight Λ. Consider the trace of the Casimir[Eq. (9.1.26)] within the possibly degenerate weight space associated to M ,

Tr

[ĈΛ2

]

M=nM 〈Λ, Λ + 2ρW〉2

=nM 〈M, M〉2 +∑

α>0

TrM

[Êα Ê−α + Ê−α Êα

]

(eα, e−α)2,

(9.2.1)

where nM is the dimension of the weight space labeled by M . In order to evaluate the second term, we consider the su(2) generatedby

[Ĥ (2)α , ʱα] = ± 〈α, α〉2 ʱα,[Êα, Ê−α] = (eα, e−α)2 Ĥ

(2)

α .(9.2.2)

It is useful to rewrite this in terms of standard su(2) generators,

[T̂z, T̂±] = ±T̂±, [T̂+, T̂−] = 2T̂z. (9.2.3)

Let

T̂± ≡ Aʱα, T̂z ≡ B Ĥα, A =√

2

〈α, α〉2(eα, e−α)2, B =

1

〈α, α〉2. (9.2.4)

For the particular root α, we have the su(2) Casimir in Eq. (9.1.1),

1

2

(T̂+T̂− + T̂−T̂+

)+ T̂zT̂z =

A2

2

[Êα Ê−α + Ê−α Êα

]+ B2 Ĥ2α. (9.2.5)

Now, each state in the nM -fold degenerate weight space of M belongs to some irreducible representation of this su(2) with highestweight t ∈ Z/2, Casimir eigenvalue t(t + 1). We choose a basis within the weight space such that each weight vector belongs toa distinct α-generated su(2) representation. The weight space may contain multiple distinguishable copies of the same irreduciblerepresentation and associated highest weight t. It may also possess one or more isolated states, i.e. states with weight M that cannotbe raised or lowered by ʱα = T̂±/A.

Consider the state in the weight space of M that sits in the α-generated su(2) with highest weight t. Denote this state bythe weight vector φ(M)t . Suppose that φ

(M+Kα)

t is the associated highest weight:

T̂z φ(M+Kα)

t = t φ(M+Kα)

t = B〈α, M + Kα〉2 φ(M+Kα)t ⇒ t = B〈α, M + Kα〉2. (9.2.6)

8

Then

[Êα Ê−α + Ê−α Êα

]φ(M)t =

{−2B

2

A2〈M, α〉22 +

2B

A2〈α, M + Kα〉2 +

2B2

A2[〈α, M〉22 + K2〈α, α〉22 + 2K〈M, α〉2〈α, α〉2

]}φ(M)t

=2B

A2

[(2K + 1) 〈M, α〉2 + K(K + 1) 〈α, α〉2

]φ(M)t . (9.2.7)

Next we note that the number of α-generated su(2) representations with label K marking the highest weight is

nM+αK − nM+α(K+1). (9.2.8)

The first term is the degeneracy of the weight space for the presumed highest weight. The second term is the degeneracy of theweight space one step above the highest weight; taking the difference eliminates the contributions of degenerate states with weightM that originate by lowering from a higher highest weight state than that assumed. To get the total contribution from the weightspace trace on the right-hand side of Eq. (9.2.1), we sum over all possible values of K. I.e., we sum over all possible highest weightstates that can give M as a descendant:

∑

α>0

TrM

[Êα Ê−α + Ê−α Êα

]

(eα, e−α)2=

∑

α>0

∑

K≥0

[nM+αK − nM+α(K+1)

] 2BA2(eα, e−α)2

[(2K + 1) 〈M, α〉2 + K(K + 1) 〈α, α〉2

]

=∑

α>0

∑

K≥0

[nM+αK − nM+α(K+1)

] [(2K + 1) 〈M, α〉2 + K(K + 1) 〈α, α〉2

]

=∑

α>0

nM 〈M, α〉2 +∑

K≥1

nM+αK

[2 〈M, α〉2 + 2K 〈α, α〉2

]

=

nM 〈M, 2ρW〉2 +∑

α>0

∑

K≥1

2nM+αK

[〈M, α〉2 + K 〈α, α〉2

]

= nM [〈Λ, Λ + 2ρW〉2 − 〈M, M〉2] . (9.2.9)

Solving for nM leads to

nM =

∑

α>0

∑

K≥1

2 nM+αK 〈M + Kα, α〉2

〈Λ + M + 2ρW, Λ − M〉2,

Freudenthal’s recursion formulafor the dimension of weight space M .

(9.2.10)

An algorithm to evaluate Eq. (9.2.10) is the following:

• Start with the highest weight of the representation Λ. This is guaranteed to be non-degenerate, so nΛ = 1.

• Proceed to each weight that differs from Λ by one root. To compute the inner products in the numerator and denominatorof Eq. (9.2.10), one can express all weights in terms of Dynkin coefficients and use the quadratic form matrix [Eq. (8.6.2)].Alternatively, to compute 〈Λ, Λ′〉 one can express

Λ =

n∑

i=1

Λiωi, Λ′ =

n∑

j=1

κ(Λ′)

j αj, (9.2.11)

so that

〈Λ, Λ′〉 =n∑

i=1

Λi κ(Λ′)

i

|αi|222

. (9.2.12)

Here we have exploited the orthonormality of the fundamental weights and simple coroots; the factor of |αi|2/2 appears becausewe have expanded Λ′ in terms of simple roots. Note that for a simply-laced algebra, |αi|22 = 2.

9

Figure 9.1: (2,1) representation of su(3). There are 15 states in this representation, but only 12 distinct weights. Each of the weightsin the inner triangle are doubly degenerate.

We will illustrate Eq. (9.2.10) for the (2,1) representation of the simply-laced algebra su(3), Fig. 9.1.

1. The highest weight state has n2,1 = 1.

2. Consider M = (0, 2). It has

Λ + M + 2ρW = (2, 1) + (0, 2) + (2, 2) = (4, 5), Λ − M = α1,⇒ 〈Λ + M + 2ρW, Λ − M〉2 = 4,

K = 1 : 2 nM+α1 〈M + α1, α1〉2 = 2 (0 + 2) = 4,(9.2.13)

so that n0,2 = 1.

3. M = (−2, 3).

Λ + M + 2ρW = (2, 1) + (−2, 3) + (2, 2) = (2, 6), Λ − M = 2α1,⇒ 〈Λ + M + 2ρW, Λ − M〉2 = 4,

K = 1 : 2 nM+α1 〈M + α1, α1〉2 = 0,K = 2 : 2 nM+2α1 〈M + 2α1, α1〉2 = 2 (−2 + 4) = 4,

(9.2.14)

so that n−2,3 = 1.

4. M = (3,−1).

Λ + M + 2ρW = (2, 1) + (3,−1) + (2, 2) = (7, 2), Λ − M = α2,⇒ 〈Λ + M + 2ρW, Λ − M〉2 = 2,

K = 1 : 2 nM+α2 〈M + α2, α2〉2 = 2 (−1 + 2) = 2,(9.2.15)

so that n3,−1 = 1.

5. M = (1, 0).

Λ + M + 2ρW = (2, 1) + (1, 0) + (2, 2) = (5, 3), Λ − M = α1 + α2,⇒ 〈Λ + M + 2ρW, Λ − M〉2 = 8,

α1, K = 1 : 2 nM+α1 〈M + α1, α1〉2 = 2(2 + 1) = 6,α2, K = 1 : 2 nM+α2 〈M + α2, α2〉2 = 2(2) = 4,

α1 + α2, K = 1 : 2 nM+α1+α2 〈M + α1 + α2, α1 + α2〉2 = 2(3) = 6,

(9.2.16)

so that n1,0 = (6 + 4 + 6)/8 = 2.

10

j = 1

j = 1

j = 3/2

j = 1/2

j = 1/2

j = 0

Figure 9.2: The same (2,1) representation of su(3) as shown in Fig. 9.1. Here we depict the decomposition into irreducible su(2)representations generated by ʱα1 , i.e. the decomposition into parallel strings consisting of weights separated by simple root α1moves. The doubly-degenerate weights arise due to overlapping representations.

We could proceed to evaluate the remaining weight space dimensions, but it is much easier to simply infer them using Weyl reflections,see Fig. 9.3. We conclude that the weights (1,0), (-1,1), and (0,-1) are each doubly-degenerate, while each weight in the “outer layer”(see Fig. 9.1) has a one-dimensional weight space. This gives a total of 12 + 3 = 15 weights, in agreement with Eq. (9.1.43).

The partitioning of the (2,1) representation into different and sometimes overlapping su(2) representations generated by ʱα1is depicted in Fig. 9.2.

• Exercise: Determine the weight tree and the dimension of all weight spaces for the su(3) representation (2,2), which has 27 totalstates [Eq. (9.1.46)]. Is there a pattern to the degeneracies in terms of the weight geometry?

• Exercise: Use Freudenthal’s formula to show that all states of the fundamental spinor representation (0,0,1) of so(7) correspondto distinct weights, i.e. that all weight spaces are one-dimensional. The weight tree (geometry) is shown in Fig. 8.9 (Fig. 8.10).

9.3 Weyl group

• • • Throughout the remainder of the course, unless stated otherwise we will employ the “standard normalization”defined by Eq. (9.1.17). We will drop explicit “2” subscripts to lighten the notation.

The Weyl reflection Sα along a root α was introduced in Sec. 5.2.1, in the context of root system geometries. The actionof Sα upon a generic weight M in an irreducible representation is given by [Eq. (5.2.3)]

Sα M = M − 〈M, α∨〉α, S2α = 1, Weyl reflection ≡ Sα. (9.3.1)

With respect to the su(2) generated by {ʱα, Ĥα}, an irreducible representation decomposes into parallel strings of weights of varyinglengths. Applying the Weyl reflection interchanges the weights along these strings. Weyl reflections for su(3) are depicted in Fig. 9.3.A simple Weyl reflection Sαi gives

Sαi M = M − Λ(M)i αi, Simple Weyl reflection, (9.3.2)

where Λ(M)i is the corresponding Dynkin coefficient. Acting on another simple root gives

Sαi αj = αj − Aji αi, (9.3.3)

where Aji ≡ 〈αj, α∨i 〉 is the Cartan Matrix [Eq. (5.3.8)].We prove the following facts about Weyl reflections.

1. Invariance of the scalar product.

〈Sα M, M ′〉 = 〈M − 〈M, α∨〉α, M ′〉 = 〈M, M ′〉 − 〈M, α∨〉〈M ′, α〉 = 〈M, Sα M ′〉. (9.3.4)

11

Figure 9.3: Weyl reflections S1,2,3 for su(3).

2. The simple Weyl reflection Sαi permutes positive roots, except for αi → −αi. Consider a positive root β ∈ ∆+ [Eq. (5.3.7)]

β ≡n∑

i=1

κ(β)i αi, κ(β)

i ∈ N0. (9.3.5)

The action of a simple root Weyl reflection is

Sαi β =

n∑

j 6=i

κ(β)j αj + αi

n∑

j 6=i

κ(β)j |Aji| − κ(β)i

. (9.3.6)

Now, assuming that the positive root β 6= αi, then at least one κ(β)j 6=i > 0. But recall that in the “root-building” application ofthe MWDF [Eqs. (5.3.26) and (7.1.24)], each positive root obtains by adding some combination of simple roots to a particularsimple root. It means that if one expansion coefficient in Eq. (9.3.6) is positive, then so are all others. Therefore Sαi β is apositive root.

Using Weyl reflections, we can prove the equivalence of the two expressions for the Weyl vector ρW in Eq. (9.1.9). Let

σ ≡ 12

∑

α>0

α, ρ ≡n∑

j=1

ωj. (9.3.7)

Since a simple Weyl reflection Sαi permutes the positive roots except αi,

Sαi σ =1

2

∑

α>0α 6=αi

α − 12αi = σ − αi. (9.3.8)

Then

〈Sαi σ, α∨i 〉 = 〈σ, α∨i 〉 − 2 = 〈σ, Sαi α∨i 〉 = −〈σ, α∨i 〉, (9.3.9)

or

〈σ, α∨i 〉 = Λi = 1, i ∈ {1, 2, . . . , n}. (9.3.10)

In this equation {Λi} denote the Dynkin coefficients of σ. We conclude that σ = ρ = ρW in Eq. (9.3.7).

The product of all Weyl reflections forms the Weyl group W , which is a subgroup of the isometry group for the root system.Any weight geometry associated to an irreducible representation must be left invariant under a Weyl group transformation. Notethat the Weyl group contains more than Weyl reflections, since the product of two reflections is equivalent to a rotation. The productof an even (odd) even number of Weyl reflections is a “proper rotation” (an “improper rotation”) with determinant equal to +1 (−1)(thinking of Sα as an orthogonal transformation on n-dimensional vectors in H

∗0 , the space of distinct weights).

12

We can choose a “basis” for the Weyl group by considering compositions of simple root Weyl reflections {Sαj}, 1 ≤ j ≤ n(and n denotes the rank of the algebra). Every element w ∈ W can be written as a product of simple Weyl reflections,

w = Sαi1 Sαi2 × · · · × Sαip . (9.3.11)

The Weyl group is important because it relates weights based on point group symmetries of the representation geometry.In module 8, we mainly stressed tools that do not make this geometry manifest. We focused on expanding weights in terms ofthe fundamental weights {ωi} with Dynkin expansion coefficients, the MWDF algorithm for constructing the weight tree, Youngtableaux, etc. The geometry of a representation is instead encoded in the expansion of a generic weight in terms of simple roots andthe geometry of the root system. Weyl group transformations algebraically encode this geometrical information.

An obvious but important statement is the following. Given the set of simple roots, the entire root geometry can beconstructed by acting on simple roots with elements of the Weyl group. In other words,

∆ ={

w α1, w α2, . . . , w αn

∣∣∣w ∈ W}

. (9.3.12)

Here ∆ denotes the set of all roots. Evidently any set {w αi} for a particular, fixed Weyl element w could serve as a basis of simpleroots. Thus we can use the Weyl group to change the basis for the simple roots.

Another obvious implication is that

nM = n(w M), w ∈ W. (9.3.13)

Here nM is the dimension of the weight space for weight M in some irreducible representation [as appears in Freudenthal’s formulaEq. (9.2.10)] and w is a generic Weyl group element.

The set of weights that can be obtained by acting with all possible Weyl group operations on some particular weight M iscalled the Weyl orbit of M . We can choose one particular weight in this orbit with non-negative Dynkin coefficients to representthe entire orbit. Proof: Suppose that M ′ is the highest weight in the orbit, and suppose that it has a negative Dynkin coefficient:Λi = 〈M ′, α∨i 〉 < 0. But then

Sαi M′ = M ′ − 〈M ′, α∨i 〉αi = M ′ + |Λi|αi (9.3.14)

is by definition (lexicographic ordering, Sec. 5.3.1) greater than M ′ (contradiction). A weight with non-negative Dynkin coefficientsis said to be dominant.

9.4 Weyl’s character formula: derivation

9.4.1 Generating functions

In the following, we will need to view H∗0 (the space of roots with real coefficients) in two different ways:

1. As a linear vector space of dimension n (where n denotes the rank), with the linearly independent simple root basis. A genericelement is a “vector” (weight)

β =

n∑

j=1

κ(β)j αj, 〈αi, αj〉 = Aij = Aij|αj|2

2. (9.4.1)

2. As an infinite dimensional Hilbert space, where |β〉 corresponding to a generic weight is itself a basis element. The inner productbetween two such states is

〈β |λ〉 = δ(n)(β − λ).A generic state is a linear function,

|f〉 =∫

dnβ f(β) |β〉 . (9.4.2)

We can view the action of a Weyl group transformation w ∈ W in these two different ways:1. As an orthogonal transformation on an n-component vector,

w β =∑

j

κ(β)j (w αj), (9.4.3)

where w αj is the transformed basis vector.

13

2. As a unitary Hilbert space operator Ûw such that

Ûw |β〉 = |w β〉 . (9.4.4)

The action on a generic function is

Ûw |f〉 =∫

dnβ f(β) |w β〉 =∫

dnβ f(w−1β) |β〉 , (9.4.5)

so that

(w f) (β) ≡ 〈β| Ûw |f〉 = f(w−1β). (9.4.6)

Now we define two different functions on H∗0 :

χ(β) = 〈β |χ〉 ≡∑

M

nM e〈M,β〉, Character of a representation.

Q(β) = 〈β |Q〉 ≡∏

α>0

[e

12 〈α,β〉 − e− 12 〈α,β〉

], (Auxiliary) Q-function.

(9.4.7a)

(9.4.7b)

The character function χ(β) involves a sum over all weights {M} of an irreducible representation; nM is the weight space dimension[Eq. (9.2.10)]. The Q-function involves a product over the positive roots.

The character formula is invariant under a Weyl transformation:

(w χ) (β) = χ(w−1β) =∑

M

nM e〈w M,β〉 =

∑

M

nw−1 M e〈M,β〉 = χ(β), (9.4.8)

where we have used Eq. (9.3.13).Let Ûαi implement a simple Weyl reflection [Eq. (9.3.2)],

Ûαi |β〉 = |Sαiβ〉 . (9.4.9)

Then

(SαiQ) (β) = 〈β| Ûαi |Q〉 =∏

α>0

[e

12 〈Sαiα,β〉 − e− 12 〈Sαiα,β〉

]

=∏

α>0α 6=αi

[e

12 〈α,β〉 − e− 12 〈α,β〉

] [e−

12 〈αi,β〉 − e 12 〈αi,β〉

]= −Q(β). (9.4.10)

Here we have used the fact that Sαi permutes the positive roots except for αi → −αi. More generally, can show that

(w Q) (β) = Q(w−1β) = (det w) Q(β), w ∈ W. (9.4.11)

Here det w is the determinant of the Weyl group element w, represented as an n × n matrix acting on the linear vector space H∗0spanned by {αi}. This determinant is guaranteed to be equal to plus or minus one for a proper or improper rotation, respectively.Eq. (9.4.11) implies that Q(β) is an “alternating” function on weight space. Equivalently, |Q〉 is an eigenvector of a generic Weyltransformation Ûw with eigenvalue detw,

Ûw |Q〉 = (det w) |Q〉 . (9.4.12)

Define

Ûσ ≡∑

w∈W

(detw) Ûw , “Alternating” (and un-normalized) projector. (9.4.13)

This has the properties

Ûw Ûσ = Ûσ Ûw = (detw) Ûσ , (9.4.14a)

(σ f) (β) ≡ 〈β| Ûσ |f〉 =∑

w∈W

(det w) f(w−1β), (9.4.14b)

Û2σ = NW Ûσ, (9.4.14c)

14

where NW is the order (number of elements) of the Weyl group.Let us attempt a decomposition for Q(β) using Ûσ:

Q(β) ≡ (σ R) (β), (9.4.15)

where R(β) is some other function. Eq. (9.4.14a) guarantees that Q(β) constructed in this way is alternating. If we expand theproduct in Eq. (9.4.7b), we will get terms of the form exp

[〈∑α>0 (±1)

mα α2 , β

〉], where mα ∈ {0, 1}. The term with all mα = 0 or

all mα = 1 is exp (〈±ρW, β〉) [Eq. (9.1.9)]. We therefore assume an expansion of the form

R(β) ≡ c0 fρW (β) +∑

α>0

cα fρW−α(β), fM (β) ≡ exp[〈M, β〉

]. (9.4.16)

Note that

(σ fM ) (β) =∑

w∈W

(detw) exp[〈wM, β〉

](9.4.17a)

(σ fw M ) (β) = (det w) (σ fM ) (β). (9.4.17b)

Eq. (9.4.17b) implies that there is no point including different weights related by Weyl transformations in the expansion Eq. (9.4.16),since their contributions to Q(β) are equivalent up to a sign.

We can therefore restrict the sum in Eq. (9.4.16) to weights M = ρW−α with non-negative Dynkin coefficients [c.f. Eq. (9.3.14)and the surrounding discussion]. In fact, (σ fM ) (β) = 0 for any weight M with at least one vanishing Dynkin label.

Proof:

• Suppose the Dynkin coefficient Λ(M)i = 0. Then [Eq. (9.3.2)]

SαiM = M.

• The function fM (β) is invariant,

(Sαi fM ) (β) = fM (Sαi β) = fSαi M(β) = fM (β) ⇒(σ fSαiM

)(β) = (σ fM ) (β).

• But then Eq. (9.4.17b) implies that

(σ Sαi fM ) (β) =(σ fSαiM

)(β) = − (σ fM ) (β).

Therefore (σ fM ) (β) = 0.

Now, the Dynkin coefficients of the Weyl vector ρW are (1, 1, . . . , 1); therefore, any weight of the form M = ρW − α is guaranteed tohave a negative or zero Dynkin coefficient. We conclude that R(β) in Eq. (9.4.16) reduces to

R(β) = c0 exp [〈ρW, β〉] . (9.4.18)

Then

Q(β) = c0∑

w∈W

(det w) e〈w ρW ,β〉 = c0 e〈ρW,β〉 + (terms involving Weyl transformations of ρW). (9.4.19)

If we compare this to Eq. (9.4.7b), we conclude that c0 = 1; therefore

Q(β) = (σ fρW ) (β) =∑

w∈W

(det w) e〈w ρW ,β〉 =∑

w∈W

(detw) e〈ρW,w β〉. (9.4.20)

The last equality holds because det w−1 = det w for an orthogonal Weyl group transformation w.

15

9.4.2 Freudenthal redux

Freudenthal’s formula [Eq. (9.2.10)] can be rewritten as

[〈Λ, Λ〉 − 〈M, M〉 + 〈2ρW, Λ− M〉]nM =2∑

α>0

∑

K≥1

nM+αK 〈M + Kα, α〉. (9.4.21)

It is useful to recast this in slightly different form.Claim:

∞∑

K=−∞

nM+αK 〈M + Kα, α〉 = 0. (9.4.22)

In this equation, it is understood that nM+αK = 0 for weights in the chain generated by ʱα that do not belong to the highestweight representation Λ.Proof: The MWDF Eq. (5.1.23) implies that

∞∑

K=−∞

nM+αK 〈M + Kα, α∨〉 =∞∑

K=−∞

nM+αK [(m− p)M ;α + 2K]

=nM+pα [(m − p) + 2p] + nM−mα [(m − p) − 2m]+

nM+(p−1)α [(m − p) + 2(p − 1)] + nM−(m−1)α [(m − p) − 2(m − 1)]+ . . .

=nM+pα [2(m − p) + 2p − 2m]+

nM+(p−1)α [2(m − p) + 2(p − 1) − 2(m − 1)] + . . .=0. (9.4.23)

Here we have used invariance of the weight space dimensions under the Sα Weyl reflection. Thus Eq. (9.4.21) can be written as

[〈Λ, Λ〉 − 〈M, M〉 + 〈2ρW, Λ− M〉]nM =∑

α>0

∑

K≥1

nM+αK 〈M + Kα, α〉 −∑

α>0

nM 〈M, α〉 −∑

α>0

∑

K≤−1

nM+αK 〈M + Kα, α〉

=∑

α>0

∑

K≥1

nM+αK 〈M + Kα, α〉 +∑

α

then

∂p

∂(βα)pe〈M,β〉 = 〈M, α〉pe〈M,β〉 ⇒

∑

α∈∆

∂2

∂(βα)2e〈M,β〉 =

∑

α∈∆

〈M, α〉〈α, M〉e〈M,β〉. (9.4.28)

Now, recall that the scalar product between two weights is defined via the Killing form:

〈M, N〉 = (hM , hN).

The Killing form can be evaluated by summing over all elements of the Lie algebra [Eq. (4.2.2)],

(x, y) =∑

αi

[x, [y, hαi ]]|hαi +∑

α 6=0

[x, [y, eα]]|eα , (9.4.29)

where O|x means take the coefficient of the Lie algebra x. Thus

〈M, N〉 = (hM , hN) =∑

α 6=0

α(hM )α(hN) =∑

α 6=0

〈M, α〉〈α, N〉. (9.4.30)

Therefore Eq. (9.4.28) implies that

∑

α∈∆

∂2

∂(βα)2e〈M,β〉 = 〈M, M〉 e〈M,β〉, (9.4.31)

and Eq. (9.4.26) becomes

[〈Λ + ρW, Λ + ρW〉 − 〈ρW, ρW〉 −

∑

α∈∆

∂2

∂(βα)2

]χ(β) =

∑

α 6=0

∑

M

IM,α(β), (9.4.32a)

IM,α(β) ≡∑

K≥0

nM+αK 〈M + Kα, α〉 e〈M,β〉. (9.4.32b)

The left-hand side of Eq. (9.4.32a) is a differential equation for the character function χ(β), viewed as a function of the “coordinates”{βα} in Eq. (9.4.27). We want to express the right-hand side of Eq. (9.2.10) in terms of the generating functions χ(β) and Q(β),Eq. (9.4.7).

Consider IM,α(β), Eq. (9.4.32b). For a fixed weight M in the weight string generated by ʱα, we must determine thecontribution of each spin-j representation of this su(2) [Eq. (9.2.2)] to IM,α(β). The spin-j representation has an odd (j ∈ Z) or even[j ∈ (2Z + 1)/2] number of weights. Let M0 locate the center of this representation. For integer j (2j + 1 odd), M0 is the centerweight; for half-integer j (2j + 1 even), M0 is not itself a weight. Then a generic weight M lying somewhere in this representationcan be written as

M = M0 + qα, q ∈ {−j,−j + 1, . . . , j − 1, j}. (9.4.33)

Suppose one copy of the spin-j representation with some (undetermined) M0 contributes to IM,α(β) in Eq. (9.4.32b). The contributionis

I(j)M,α(β) ≡j−q∑

K≥0

(1) 〈M0 + (K + q)α, α〉 e〈M0+qα,β〉 = 〈α, α〉 e〈M0,β〉j−q∑

K≥0

(K + q) eq〈α,β〉 , (9.4.34)

where we’ve used the fact that 〈M0, α〉 = 0; the latter is obvious for the case where M0 is itself the center weight [Eq. (5.1.23)]. Wecan sum over the nondegenerate weights [Eq. (9.4.33)] in this spin-j representation to get

I(j)α (β) ≡∑

M

I(j)M,α(β) = 〈α, α〉 e〈M0,β〉j∑

q=−j

j−q∑

K≥0

(K + q) eq〈α,β〉 . (9.4.35)

17

Consider

[e〈α,β〉 − 1

]I(j)α (β) = 〈α, α〉 e〈M0,β〉

j∑

q=−j

j−q∑

K≥0

(K + q) e(q+1)〈α,β〉−j−1∑

q=−j−1

j−q−1∑

K≥0

(K + q + 1) e(q+1)〈α,β〉

= 〈α, α〉 e〈M0,β〉

j e(j+1)〈α,β〉 +

j−1∑

q=−j

j−q−1∑

K≥0

(K + q) e(q+1)〈α,β〉 +

j−1∑

q=−j

j e(q+1)〈α,β〉

−j−1∑

q=−j

j−q−1∑

K≥0

(K + q + 1) e(q+1)〈α,β〉 −2j∑

K≥0

(K − j) e−j〈α,β〉

= 〈α, α〉 e〈M0,β〉

j e(j+1)〈α,β〉 +j−1∑

q=−j

j e(q+1)〈α,β〉 −j−1∑

q=−j

(j − q) e(q+1)〈α,β〉

= 〈α, α〉 e〈M0,β〉j∑

q=−j

q e(q+1)〈α,β〉 .

(9.4.36)

Thus

I(j)α (β) =1[

e〈α,β〉 − 1]

j∑

q=−j

〈qα, α〉 e〈M0+qα+α,β〉 = 1[e〈α,β〉 − 1

]j∑

q=−j

〈M, α〉 e〈M+α,β〉 , (9.4.37)

where we have used Eq. (9.4.33) and the fact that 〈M0, α〉 = 0. Eq. (9.4.37) is the contribution to the right-hand side of Eq. (9.4.32a)of all weights in the spin-j su(2) generated by ʱα, with central weight M0. The key result is that Eq. (9.4.37) does notdepend explicitly on M0, but only implicitly through the (summed) weight M [Eq. (9.4.33)].

We can therefore rewrite the right-hand side of Eq. (9.4.32a) as

RHS ≡∑

α 6=0

∑

M

IM,α(β) =∑

α 6=0

∑

M0

∑

j

n(α)M0,j I(j)

α (β). (9.4.38)

For each root α, we must sum over all central weights M0 and all representations j; there is an (undetermined) degeneracy factorn(α)M0,j that can be zero, one, or more if there are multiple spin-j su(2) representations with the same central weight that contribute.At this point it looks like we have just introduced many more unknowns, but now a minor miracle ensues. Combining Eqs. (9.4.37)and (9.4.38), we get

RHS =∑

α 6=0

1[e〈α,β〉 − 1

]∑

M0

∑

j

j∑

q=−j

n(α)M0,j 〈M, α〉 e〈M+α,β〉 =

∑

α 6=0

1[e〈α,β〉 − 1

]∑

M

nM 〈M, α〉 e〈M+α,β〉, (9.4.39)

since the sum over representation centers M0, representations j, and weights q ∈ {−j, . . . , j} weighted by the degeneracy n(α)M0,j isjust a particular decomposition of the sum over all weights M (again weighted by the appropriate degeneracy).

As an example, consider the (2,1) representation of su(3) shown in Fig. 9.2. For simple root α1, the center weights {M0}enumerate the four parallel horizontal weight strings. Each horizontal string consists of one or two different spin-j representations.Summing over all M0, all j, and states within each j is equivalent to summing over all weights times the appropriate degeneracies.Note that in this case, n(α1)M0,j = 1 for all M0 and j.

9.4.4 finis

Replacing the right-hand side of Eq. (9.4.32a) with Eq. (9.4.39), we have

[〈Λ + ρW, Λ + ρW〉 − 〈ρW, ρW〉 −

∑

α∈∆

∂2

∂(βα)2

]χ(β) =

∑

α 6=0

∑

M

nM 〈M, α〉 e〈M+α,β〉[e〈α,β〉 − 1

] . (9.4.40)

18

Eq. (9.4.7b) implies that

Q2(β) =∏

α>0

[e〈α,β〉 − 1

] [1 − e−〈α,β〉

]=

∏

α 6=0

[e〈α,β〉 − 1

](−1)|∆+|, (9.4.41)

where |∆+| denotes the number of positive roots. Then

∂

∂βαlog[Q2(β)] =

∑

α′ 6=0

〈α′, α〉 e〈α′,β〉[e〈α′,β〉 − 1

] , (9.4.42)

where βα is an expansion coefficient for weight β in the overcomplete set of all roots, Eq. (9.4.27). Combining the above withEq. (9.4.28), we get

∑

α 6=0

{∂

∂βαlog[Q2(β)]

}{∂

∂βαχ(β)

}=

∑

M

nM∑

α′ 6=0

∑

α 6=0

〈α′, α〉〈α, M〉e〈M+α′,β〉[e〈α′,β〉 − 1

]

=∑

M

nM∑

α′ 6=0

〈α′, M〉e〈M+α′,β〉[e〈α′,β〉 − 1

]

=1

Q

∑

α 6=0

[∂2

∂(βα)2(Qχ) (β) − χ ∂

2

∂(βα)2Q(β) − Q ∂

2

∂(βα)2χ(β)

]. (9.4.43)

On the second line, we have used Eq. (9.4.30). Eq. (9.4.40) becomes

[〈Λ + ρW, Λ + ρW〉 − 〈ρW, ρW〉] (Qχ) (β) =∑

α 6=0

[∂2

∂(βα)2(Qχ) (β) − χ ∂

2

∂(βα)2Q(β)

]. (9.4.44)

Eq. (9.4.20) implies that

∑

α 6=0

∂2

∂(βα)2Q(β) =

∑

α 6=0

∑

w∈W

(det w) 〈w ρW, α〉〈α, w ρW〉 e〈w ρW,β〉

=∑

w∈W

(detw) 〈w ρW, w ρW〉 e〈w ρW,β〉

= 〈ρW, ρW〉Q(β). (9.4.45)

Therefore Eq. (9.4.44) becomes∑

α 6=0

∂2

∂(βα)2(Qχ) (β) = 〈Λ + ρW, Λ + ρW〉 (Qχ) (β). (9.4.46)

Note that (Qχ)(β) is an alternating function [Eqs. (9.4.8) and (9.4.11)]. Eqs. (9.4.46), (9.4.45), and (9.4.20) suggest the followingsimple ansatz (guess):

(Qχ) (β) =∑

w∈W

(det w) e〈Λ+ρW,w β〉. (9.4.47)

This obviously satisfies Eq. (9.4.46), but it is not the unique solution. The precise form of Eq. (9.4.47) can be proven following asimilar line of argument used to obtain Eq. (9.4.20) from Eqs. (9.4.15) and (9.4.16).

Combining Eqs. (9.4.47) and (9.4.20), we finally obtain

χ(β) =∑

M

nMe〈M,β〉 =

∑

w∈W

(det w) e〈Λ+ρW,w β〉

∑

w∈W

(det w) e〈ρW,w β〉, Weyl’s character formula. (9.4.48)

Eq. (9.4.48) is essentially a resummation of Freudenthal’s recursion formula Eq. (9.2.10), assisted by the Weyl group.

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9.5 Dimension and strange formulae

The Weyl character χ is a functional defined for any irreducible representation; in Eq. (9.4.48) it is evaluated for an arbitrary elementβ ∈ H∗0 . If we set

β = t ρW (9.5.1)

with t ∈ R, then using Eq. (9.4.7b)

χ (t ρW) =

∑

w∈W

(det w) e〈w ρW,t(Λ+ρW)〉

∑

w∈W

(det w) e〈w ρW ,t ρW〉=

Q [t (Λ + ρW)]

Q (t ρW)

=∏

α>0

sinh[12 〈α, t (Λ + ρW)〉

]

sinh[12 〈α, t ρW〉

] . (9.5.2)

Therefore

limt→0

χ (t ρW) =∑

M

nM = NΛ ' limt→0

∏

α>0

〈α, t (Λ + ρW)〉∏

α>0

〈α, t ρW〉, (9.5.3)

where NΛ is the dimension of the representation. We thus have the dimension formula

NΛ =∏

α>0

〈α, Λ + ρW〉〈α, ρW〉

, Dimension NΛ of highest weight representation Λ. (9.5.4)

We can obtain another formula from Eq. (9.5.2). Expanding again for small t,

χ (t ρW) '∏

α>0

〈α, (Λ + ρW)〉[1 + t

2

24 〈α, (Λ + ρW)〉2 + . . .]

〈α, ρW〉[1 + t

2

24 〈α, ρW〉2 + . . .]

= NΛ

1 +t2

48

∑

α 6=0

[〈(Λ + ρW) , α〉〈α, (Λ + ρW)〉 − 〈ρW, α〉〈α, ρW〉] + . . .

= NΛ

{1 +

t2

48[〈(Λ + ρW) , (Λ + ρW)〉 − 〈ρW, ρW〉] + . . .

}, (9.5.5)

where we have used Eq. (9.4.30). On the other hand,

χ(t ρW) =∑

M

nMe〈M,t ρW〉 =

∑

M

nM

[1 + t〈M, ρW〉 +

t2

2!〈M, ρW〉2 + . . .

](9.5.6)

The terms linear in t sum to zero due to Weyl invariance. If we take Λ = θ (the highest root, i.e. we are studying the adjointrepresentation), then the sum over M is replaced by a sum over non-zero roots (since the inner product with the zero weightspecifying the Cartan subalgebra gives zero). Therefore

χ(t ρW) = Nθ +t2

2!

∑

α 6=0

〈α, ρW〉2 + . . . = Nθ +t2

2〈ρW, ρW〉 + . . . , (9.5.7)

where we have used Eq. (9.4.30) and the non-degeneracy of the roots nα = 1 (Proposition (VI.), Sec. 5.2.2]. Eqs. (9.5.5) and (9.5.7)imply that

〈ρW, ρW〉 =Nθ24

〈θ, θ + 2ρW〉. (9.5.8)

20

Here Nθ = d is the dimension of the adjoint representation (or the Lie group). In the standard normalization we obtain

〈ρW, ρW〉2 =Nθ g

12, Freudenthal-de Vries strange formula. (9.5.9)

Here g denotes the dual Coxeter number [Eq. (9.1.21)].

9.5.1 Dimensions for generic representations of the classical algebras

There are different methods to evaluate Eq. (9.5.4) for a generic highest weight representation Λ in a particular Lie algebra. We willuse a strategy that works for all algebras and can be easily evaluated numerically.

The simplest case (unsurprisingly) is An−1 = su(n). In Sec. 7.1, we showed that the positive roots can be enumerated {αij}with 1 ≤ i < j ≤ n. These are expressed in terms of the simple roots via Eq. (7.1.11),

αij =

j−1∑

k=i

αk =

j−1∑

k=i

α∨k . (9.5.10)

Then

〈Λ + ρW, αij〉 =j−1∑

k=i

(Λk + 1) = (j − i) +j−1∑

k=i

Λk ≡ 2F(Λ; i, j). (9.5.11a)

Therefore

NΛ =

n−1∏

i=1

n∏

j>i

[F(Λ; i, j)F(0; i, j)

], Dimension formula for su(n) = An−1. (9.5.12)

• Exercise: Use Eq. (9.5.12) to show that the Λ = (p, q) representation of su(3) has Np,q given by Eq. (9.1.43).• Exercise: Use Eq. (9.5.12) to show that the Λ = ωp representation of su(n) has dimension

Np =n!

p!(n− p)! , (9.5.13)

consistent with a rank-p antisymmetric tensor [c.f. Eq. (8.2.7)].The dimension formula for sp(2n) can be similarly obtained using Eqs. (7.1.11) and (7.2.12). The result is

NΛ =

n∏

i=1

[F(Λ; i, n + 1)F(0; i, n + 1)

] n∏

j>i

{F(Λ; i, j) [F(Λ; i, j) + 2F(Λ; j, n + 1)]F(0; i, j) [F(0; i, j) + 2F(0; j, n + 1)]

}, Dimension formula for sp(2n) = Cn. (9.5.14)

• Exercise: Derive Eq. (9.5.14).• Exercise: Use Eq. (9.5.14) to verify that the fundamental representation Λ = ωp (1 ≤ p ≤ n) has dimension

Nωp = 2(n + 1 − p)(2n + 1)!

p!(2n− p + 2)! , (9.5.15)

consistent with a “traceless” fully antisymmetric rank-p tensor [Eq. (8.3.11)].We can also obtain generic dimension formulae for so(2n) = Dn and so(2n+1) = Bn, but we won’t transcribe the results here.

The most important are the dimensions of the fundamental spinor representations, which are Nωn = 2n for the ωn representation of

Bn, and Nωn−1 = Nωn = 2n−1 for the independent ωn−1 and ωn representations of Dn.

References

[1] Robert N. Cahn, Semi-Simple Lie Algebras and Their Representations (Benjamin/Cummings, Menlo Park, California, 1984).

[2] Phillipe Di Francesco, Pierre Mathieu, David Sénéchal, Conformal Field Theory (Springer-Verlag, New York, 1996).

[3] Jürgen Fuchs, Affine Lie Algebras and Quantum Groups (Cambridge University Press, Cambridge, England, 1995).

21