3. Group Representations

3.1 Representations3.2 Irreducible, Inequivalent Representations3.3 Unitary Representations3.4 Schur's Lemmas3.5 Orthonormality and Completeness Relations of Irreducible

Representation Matrices3.6 Orthonormality and Completeness Relations of Irreducible

Characters3.7 The Regular Representation3.8 Direct Product Representations, Clebsch-Gordan Coefficients

3.1. Representations

Definition 3.1: Group Representations

Let be the space of linear operators on an n-D vector space V.

Let U: G , g U(g) be a homomorphism, i.e,

U(g g') = U(g) U(g') g, g' G

Then U(G) = { U(g) | gG } is a n-D representation (rep) of G.

The rep is faithful if U is an isomorphism.

Def: Group of Operators / Linear Transformations

= Set of invertible linear operators on a linear space that is also a group

Let { ej | j = 1, 2, …, n } be a basis for the n-D vector space V.

U(g) can be realized as n n matrices D(g) according to

1

n

k jk

U g j k D g

j jeg G

k jk D g (Einstein's summation notation)

k jU g U g j U g k D g i k

k ji D g D g i ji D gg

i k i

k j jD g D g D gg D g D g D gg

D(G) forms a matrix representation of G.

Example 1: Trivial 1-D Rep for Every Group, V =

U: g U(g) = 1 gG

Example 2: Non-Trivial 1-D Rep for Group of Matrices

G = group of matrices & V = C. U(g) = det g.

Example 3: 1-D Rep for Td

U [ T(n) ] = e – i n { – < )Example 4: 2-D Rep for D2 = { e, C2, x, y } = Symm of Rectangle

1 0

0 1D e

2

1 0

0 1D C

1 0

0 1xD

1 0

0 1yD

kj j k jU g D g e e e

1

1

1

1

1

1

ni

n n j

n

n

n

D g

D g

D g

D g

D g

e e e e

Example 5: R(2) = { R(), 0 < 2 } , V = 2

1 1U e e 1 2cos sin e e

2 2U e e 1 2sin cos e e

cos

si

si

c sn

n

oD

2

1i i

i

x

x e iixe 2 x E

U x x iiU x e i

ixe j ij iD xe j

jxe

jj i

ix D x 1 1

2 2

cos sin

sin cos

x x

x x

1 1 2cos sinU U U e e e

1 2 1 2cos sin cos sin cos sin e e e e

1 2cos cos sin sin sin cos cos sin e e

1 2cos sin e e 1U e

D x x

2 2U U U e eSimilarly U U U

{ U() , 0 < 2 } is a 2-D representation of R(2) on 2

{ D() , 0 < 2 } is a 2-D matrix rep of R(2) wrt basis {e1, e2 }

D D D Also

Example 6: D3 or C3v = { E, C3, C32, σ1, σ2, σ3 } , V = 2

The 2-D matrix rep is

1 0

0 1D E

3

1 3

2 2

3 1

2 2

D C

231 3

2 2

3 1

2 2

D C

11 0

0 1D

2

1 3

2 2

3 1

2 2

D

3

1 3

2 2

3 1

2 2

D

Example 7: f = { f(x,y) = a x + b y | x,y , a,b }f is the space of complex–valued linear homogeneous functions of 2 real variables.

Let G be any one of the groups in the previous examples.

Let U be a rep of G on 2 so that

ggf f f U

g U g x x x x = {x,y} 2 & g G

This induces a rep of G on f as follows

where f f x x 1f U g

x

1

g f f f U g x x xU

This mapping is indeed a homomorphism

Let g = h k, then

1

h k hf f U k x xUU U 1 1

f U k U h x 1

f U hk x hk f xU

The ensuring 2-D matrix rep's are identical to those found earlier ( prove it ! )

f is an (scalar) example of a field.

Higher dim versions vector, spin, isospin, …

Theorem 3.1: G/H

1. If G has a non–trivial invariant subgroup H, then any rep of K = G/H is also a degenerate rep of G.

2. Conversely, if U(G) is a degenerate rep of G, then G has at least one invariant subgroup H such that U(G) defines a faithful rep of G/H.

Proof:

1. Let U be a rep of K on V so that

U: K , k U(k) = space of linear operators on V.

: G , g (g) = U(k) where g h = k K for some hH

is a homomorphism.

is a rep for G.

Mapping is many–1 since H is non-trivial.

2. Follows from Theorem 2.5Theorem 2.5:

Let : G G' be a homomorphism and Kernel = K = { g | (g) = e' }

Then K is an invariant subgroup of G and G/K G'

Example: S3 = { e, (123), (132), (23), (13), (12) }

Invariant subgroup: H = { e, (123), (321) } C3.

Factor group: S3 / H C2 = { e, a }

Non-trivial 1–D rep for C2: { e, a } {1, –1}

Induced degenerate rep for S3 :

{ e, (123), (132), (23), (13), (12) } { 1,1,1, –1,–1,–1 }

e (123) (132) (23) (13) (12)

(123) (132) e (12) (23) (13)

(132) e (123) (13) (12) (23)

(23) (13) (12) e (123) (132)

(13) (12) (23) (132) e (123)

(12) (23) (13) (123) (132) e

Corollary: All rep's, except the trivial one, of simple groups are faithful.

e.g., Cn with n prime

3.2. Irreducible, Inequivalent Representations

Definition 3.2: Equivalent Representations

Two rep's of G are equivalent if they are related by a similarity transform.

Definition 3.3: Characters

The character of gG in rep U(G) is defined as

Since trace is preserved in a similarity transform, all elements in a class have the same character.

Definition 3.4: Invariant Subspace

Let U(G) be a rep of G on V. A subspace V1 of V is invariant wrt U(G) if

U(g) | x V1 xV1 & gG

V1 is proper / minimal if it doesn't contain any non-trivial invariant subspace

g TrU g j

j U g jFor a matrix rep D(G), i i

i

g D g i

iD g

j U g j

Definition 3.5: Irreducible Representations (IR)

A rep U(G) of G on V is irreducible if there is no non-trivial invariant subspace in V wrt U(G).

Otherwise, U(G) is reducible.

A reducible rep is decomposable if the orthogonal complement of the invariant subspace is also invariant.

Example 1: D2 C2v = { e, C2, x, y } V = 2 c.f. Eg 4, §3.1

The x-axis, or span( e1), is a minimal invariant subspace wrt D2.

Ditto y-axis = span(e2).

2-D rep in Eg 4, §3.1, is decomposable.

1 0

0 1D e

2

1 0

0 1D C

1 0

0 1xD

1 0

0 1yD

Example 2: R(2) V = 2 c.f. Eg 5, §3.1

1-D invariant subspaces wrt R(2) are spanned by 1 2

1

2i e e e

1 2 1 2

1cos sin sin cos

2U i e e e e e

1 1 2cos sinU e e e 2 1 2sin cosU e e e

1 2

1cos sin cos sin

2i i i e e ie

e

Matrix rep wrt { e+, e– } is 0

0

i

i

eD

e

Example 3: D3 = { E, C3, C32, C2, C2', C2'' } V = 2

2 is minimal wrt D3

2-D rep in Problem 3.1 is an IR

Let V1 be an n1-D invariant subspace wrt a reducible U(G).

Choose basis { ej | j = 1,…,n } so that 1st n1 vectors are in V1.

1

1

nj j

i ij

U g i j D g j D g

11, ,i n

0j

iD g 1 11, , ; 1, ,i n j n n

1

2

D g D gD g

O D g

where D1(g) is n1 n1

Since ' ' ' ' '

' '

A B A B AA AB BC

O C O C O CC

D(g g') = D(g) D(g') is also upper triangular

If V2 = span { ej | j = n1+1,…,n } is also invariant,

then U(G) is decomposable & all D'(g) = O

Restricting U(G) to an invariant subspace results in a rep of lower dim.

0 0

A B x Ax

O C

0 0A O

B C y Cy

3.3. Unitary Representations

Definition 3.6: Unitary Representation

A rep U(G) of G on V is unitary if

• V is an inner product space

• U(g) are unitary g G

Unitary operators preserve inner products & thereby,

lengths & angles

V1 is invariant 1U g j g j V 11, ,j n

U(g) is unitary †0 k j k U g U g j

Orthogonal complement 0k j 1 11, , & 1, ,k n n j n

g k g j

2U g k g k V QED

Theorem 3.2: Unitarity: Reducibility Decomposability

If a unitary representation is reducible, then it is also decomposable.

Proof:

Let U(G) be the unitary reducible rep of G on inner prod space

V = span{ ei | i = 1, …, n }

Let V1 = span{ ej | j = 1, …, n1 } be the relevant invariant subspace &

V2 = span{ ek | k = n1+1, …, n } its orthogonal complement.

Theorem 3.3:

Every rep D(G) of a finite group on an inner prod space is equiv to a unitary rep.

Proof :

We shall show that S D(g) S–1 = U(g) is unitary gG if S satisfies

g G

S x S y D g x D g y

The existence of S is established by showing that ( , ) is an inner product so that S is just the relevant basis transformation.

( Proof of this is left as an exercise: see Prob 3.4 )

,x y

Proof of unitarity of U(G) is as follows:

1 1U g x U g y SD g S x SD g S y

1 1

g G

D g D g S x D g D g S y

1 1

g G

D g S x D g S y

1 1,S x S y x y g g g

Comments:

• Theorem can be extended to infinite groups for which an invariant measure can be defined, e.g., compact & semi-simple Lie groups.

• All reducible rep's of a finite group are decomposable.

The inner product space is a direct sum of the invariant subspaces.

Definition 3.7: Direct Sum Representation

Let U(G) be decomposable on V. Then

V = V1 V2 U(G) = U1(G) U2(G)

jj

j j

V V U G U G n U G

where U is an IR that occurs n times.

D(G) will be block diagonal for a properly chosen basis

3.4. Schur's Lemmas

Schur's Lemma 1:

Let U(G) be an IR of G on V, and A . Then

AU(g) = U(g)A gG A = E

Proof:

• Let U(G) be unitary, else replaced by its unitary equivalence S U(G) S–1.

• Let A be hermitian, else replaced by one of its hermitian components

A+ = ( A + A+) / 2 or A– = ( A – A+) / 2i

• Since A is hermitian, it is diagonalizable & its eigenvalues are all real.

Let | j be the eigenstate corresponding to the eigenvalue j of A :

jA j j

where { | j } can be chosen to be orthonormal:

kjk j

Label is introduced to account for possible degeneracies, viz.,

jU g j AU g j U g A j A U(g) = U(g) A

is also an eigenvector belonging to j U g j

Let n be the degree of degeneracies.

; 1, ,jV span j n is an invariant subspace wrt U(G).

U(G) is IR V has no non-trivial invariant subspaces

Vj = V & j = 1 only

A = 1 E where D(E) is n n

If A itself is hermitian, we have A = E, where = 1 is real.

Otherwise, A = ( + + i – ) E, where is the (real) eigenvalue of A

Theorem 3.4:

IR of any abelian group must be 1-D.

Proof:

Let U(G) be an IR of the Abelian G. For a given pG,

U(p) U(g) = U(g) U(p) gG

Schur‘s 1st lemma U(p) = p E pG

U(G) is equivalent to the 1-D rep { p p }

Schur's Lemma 2:

Let U(G) & U'(G) be IRs of G on V & V' , resp.

Let A: V' V be linear & satisfies

A U' (g) = U(g) A gG

Then, either

1) A = O , or

2) V V' (A is isomorphism)

& U(G) U'(G) (equivalent)

Proof:

Let R = Range A = { x V | x = A x' , x' V' }

U g x U g A x AU g x A U g x R g G

R is an invariant subspace of V wrt U(G)

U(G) is IR

1) Either R = { 0 } & hence A = O

2) or R = V so that map A is onto

Proof is done if for 2), A is also 1-1

Let N' = Null ( V' ) = { x' V' | A x' = 0 }

AU g x U g A x

0 0U g &N g G x

i.e., N' is an invariant subspace of V' wrt U’(G)

N U g x N x

U' (G) is IR

1) Either N' = V' & hence A = O

2) or N' = { 0 } so that map A is 1–1 since

0 0A x y x y

A x A y x y

Thus, A O is an isomorphism &

A U' (g) = U(g) A U(g) = A U' (g) A–1

3.5.Orthonormality & Completeness Relations of IR Matrices

Notations:

nG order of the group G

, labels for inequivalent IRs of G

n dimension of the -rep

D(g) matrix version of gG in the -rep wrt an orthonormal basis

j character of elements of class j in the -rep

nj number of elements in the class j

nC number of classes in G

Theorem 3.5a : Orthonormality of IR Matrices

1 i k i km jmj

g GG

nD g D g

n

1

1 1 11

G

G

k n

k n

g g g

g g g

kmG

nD g

n is an nG–D orthonormal vector for given ( , k, m )

dual

Proof to be given after some examples

Theorem 3.5: Orthonormality of Unitary IR Matrices

*i k ik

jmj mg GG

nD g D g

n

† j k k ji mi m

g GG

nD g D g

n

kmG

nD g

n is an nG–D orthonormal vector for given ( , k, m )

1

* * *

1

G

G

k n

k n

g g g

g g g

dual

Example 1: C2 = { e, a } nG = 2

Identity rep: 1, 1, 1de a

1-D rep orthonormal to d1 : 2, 1, 1de a

C2 e a

1 1 1

2 1 –1

Example 2: D2 = { e, a = a–1, b = b–1, c = a b } nG = 4

e a b c

a e c b

b c e a

c b a e

Identity rep: 1, , , 1, 1,1, 1de a b c

Invariant subgroup { e, a } C2

Factor group D2 / C2 C2

D2 e a b c

1 1 1 1 1

2 1 1 –1 –1

3 1 –1 1 –1

4 1 –1 –1 1

e a

a e

For Abelian groups, all IRs are 1–D: *1

GG g

d g d gn

Example 3: Td = { T(n) | n } nG = c.f. Chap 1

i nd T n e

Td Abelian All IRs are 1–D

Orthonormality:

* i n in

n n

d T n d T n e e

23 3 1 2 3

1

2

3 11

3 1

3 2

3

3

2

1

22

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 11 1

2 2 2 2

3 3 3 30 0

2 2 2 2

3 3 3 30 0

2 2 2 21 1 1 1

1 12 2 2 2

v E C CC

23 3 1 2 3

1

2

3

3

3 3 3 30 02 2

1 1 1 1 1 1

1 1 1 1 1 1

0 1 0 13 1 3 1 3 1 3 1

1 1 1 1

2 2 2 2

1 12 2 2 2

2

2 2

2 2 2

vC E C C

Example 4: C3v = { E, C3, C32, σ1, σ2, σ3 } nG = 6

*

g

j k k jGi mi m

nD g D g

n

Proof of Theorem 3.5a:

Let X be any nn matrix and 1X

g

M D g X D g

1 1 1

gXD p M D p D p D g X D g D p

1

g

D g X D g

g g p

XM p G

Schur's lemmas:

Either & MX = 0

or = & MX = cX E

Let ik k k im m j mj

X X

1km

k km mX

g

M M D g X D g

,

1i li nk km mj

gl jn

l n

M D g X D g 1 i k

mg

jD g D g

1

g

i k k im jm j

D g D g c

Take trace on both sides 1 i kkm m i

g i

c n D g D g

kG mn

k kGm m

nc

n

QED

Corollary 1: 2Gn n

Proof:

kmG

nD g

n is an nG–D orthonormal vector for given ( , k, m )

Since for a given rep , there are n2 choices of ( k, m ).

2n is the number of orthonormal vectors in this nG–D space

QED

Comments:

• Corollary allows meaningful search for all inequivalent IRs for finite / compact / semi-simple groups.

• Next theorem shows that

2Gn n

Theorem 3.6: Completeness of IR Matrices

1)2

Gn n

2) †

, ,

k j ggj k

Gk j

nD g D g

n

*

, ,

kk ggj j

j kG

nD g D g

n

Proof of 1) is deferred to § 3.7.

Given 1) , 2) is just the completeness relation of nG orthonormal vectors in an nG–D vector space.

Comments:

• If G is Abelian, n = 1 nG inequivalent IRs.

• Application of Thm 3.5-6 to infinite groups requires existence of an invariant measure to replace group sum with an integral

• Thm 3.5-6 are basis-dependent; character versions of them are not.

Example : C3v = { E, C3, C32, σ1, σ2, σ3 } nG = 6

23 3 1 2 3

1

2

3 11

3 1

3 2

3

3

2

1

22

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 11 1

2 2 2 2

3 3 3 30 0

2 2 2 2

3 3 3 30 0

2 2 2 21 1 1 1

1 12 2 2 2

v E C CC

*

g

i k ikGjmj m

nD g D g

n

*

, ,

kk

G g gj jj k

n D g D g n

23 3 1 2 3

1

2

3

11

3

12

3

21

3

22

3

1 1 1 1 1 1

6 6 6 6 6 6 61 1 1 1 1 1

6 6 6 6 6 6 6

1 1 1 1 1 1

3 3 2 3 2 3 3 2 3 2 3

1 1 1 10 0

2 2 2 23

1 1 1 10 0

2 2 2 23

1 1 1 1 1 1

3 3 2 3 2 3 3 2 3 2 3

v EC C C

*

i k i kjmj

G Ggm

n nD g D g

n n

*

, ,

k k

g gj jj k G G

n nD g D g

n n

3.6. Orthonormality and Completeness Relations of Irreducible Characters

Lemma: Sum over a Class

Let U(G) be an IR of G . Then j

jj

h

nU h E

n

Proof:

j

jh

A U h

Let

then 1 1

j

jh

U g A U g U g U h U g

jh

U h

1h ghg

jA g G

Schur's 1st lemma: j jA c E

Take trace : . . .j

j jh

L H S h n

. . . jR H S c n j

j j

nc

n

QED

Theorem 3.7: Orthonormality & Completeness of

*

j

jj j Gn n

*k kj j G jn n

(Orthonormal

)(Complete)

Proof: †

g

j k k jG i mi m

n D g D g n

†

, ,g

i k k iG i

ikk

ii

k k

n D g D g n

*

gGn g g n n

*

j

jj j Gn n

†

, ,

jk

G gj k

gj kn D g D g n

, ,

*

,

. . .m i j

k i jm im j k

kg g

n nL H S n

n n

*im i mn n

,

. . .m i

iG m i

g g

R H S n n

*i im m G mn n

QED

1, , Cj n

Define the normalized character as jj j

G

n

n

* j

jj

Orthonormality†k k

j j

Completeness

Corollary: ( for finite groups )

Number of inequiv IRs = Number of classes = nC

Since j = 1, …, nC , { j } is an nCnC matrix ( character table ).

Example: Abelian groups

Each group element forms a class by itself and all IRs are 1–D,

i.e., D(gj) = j

Tables of D(G) are also tables of characters

Note: Tables of characters are independent of basis.

Example: S3 = { e, (123), (132), (23), (13), (12) }

3 classes 3 inequiv IRs

Identity rep: 1 = ( 1, 1, 1 )

e (123) (132) (23) (13) (12)

(123) (132) e (12) (23) (13)

(132) e (123) (13) (12) (23)

(23) (13) (12) e (123) (132)

(13) (12) (23) (132) e (123)

(12) (23) (13) (123) (132) e

1–D rep 2 = ( 1, a, b ). Orthonormality 1 2 3 0a b

2 = ( 1, 1, –1 )2 2 21 1 6d Dim d of

3 :2d

3 = ( 2, a, b ). Orthonormality 2 2 3 0a b 3 = ( 2, –1, 0 )

S3 e 2 (123) 3 (23)

1 1 1 1

2 1 1 –1

3 2 –1 0

e n

3

1

2

3

2 123 3 23

1 1 1

3 26

1 1 1

3 26

2 10

36

S e

Theorem 3.8:

U G a U G

† j

j

jj

G

na

n † jj

j †

j ja

Proof: Take trace of U G a U G

†

,

j jjj

j

jj

Gj G

n na

n n

a

Example: C2 = { e = a2, a }

D(G):

1 0

0 1e

1 0

0 1a

= ( 2, 0 )

C2 e a

1 1 1

2 1 –1

2 01 2a b

11 2 1 0

2a 1 1

1 2 1 02

b 1

1 2U G U G U G D(G) D'(G):

1 0

0 1e

0 1

1 0a

Prob 3.6

Example: Vibration of NH3

C3v e 2C3 3 v

A1 1 1 1 z

A2 1 1 –1 Rz

E 2 –1 0 (x,y), (Rx, Ry)

Nxy 2 –1 0 E

Nz 1 1 1 A1

Hxy 6 0 0 A1+ A2 + 2E

Hz 3 0 1 A1 + E

NH3 12 0 2 3A1 + A2

+ 4E

Translation:

A1+ E

Rotation:

A2 + E

Vibration:

2A1+ 2E

Example: Electronic states of NH3

C3v e 2C3 3 v

A1 1 1 1 z

A2 1 1 –1 Rz

E 2 –1 0 (x,y), (Rx, Ry)

Ns 1 1 1 A1

Np 3 0 1 A1 + E

Hs 3 0 1 A1 + E

NH3 7 1 3 3A1 + 2E

Bonds 3 0 1 A1 + E

N: 1s2 2s22p3

H: 1s

Theorem 3.9: Condition for Irreducibility

U(G) is IR 2

1

Cn

j j Gj

n n

i.e.,† 1

Proof: Let a

*

,

a a

,

*a a

2

1

Cn

a

If U(G) = U(G), then1

0a

otherwise

† 1

Conversely † 1 2

1

1Cn

a

so that 1

0a

otherwise

for some

i.e., U(G) = U(G)

Character tables of all crystallographic point-groups are given in most texts

3.7. The Regular Representation

More details concerning group algebra are in Appendix III. See also Chap 5

Notations for group multiplications in finite G = { gj ; j = 1, …, nG } :

i j kg g g mi j m i jg g g 1

0m i jm

i j

g g g

otherwise

Theorem 3.10: Regular Representation

ab c mm abab g 1

0mm

ab

g ab c

otherwise

1 , ,kR R k k

i j i j i GjD G D g i n is an nG–D matrix rep of G

Proof: Let a b = c , where a, b, c G

k k

a b cj j

mj m b jabg ag k m

k am b jg kj k c jcg g

a b = c k m kam b j c j a b c

QED

Comments:

• Theorem 3.10 is just a version of the Cayley's theorem ( Thm 2.1 ).

1

1a n

n

na G p S

a a

Cayley:

Reg rep:

aa G

withia ig ag

i ia a jj

withj

ia i

a j P

where Pa is the nn matrix rep of pa.

• Alternative proof for Theorem 3.10:

k k ma b am b jj

b j

ka

m j

k ma b k

ab jj

kab kab j

ia ig ag i

k ka a i

1 1

1 1ab a b

n n

n np p p

a a b b

1

1

1

1

n

n

nb b

b b n

b ba a

1

1

nb b

n

a a

where

Theorem 3.11: Decomposition of the regular rep R

Proof 1:

R n

R ja a a

jjTr j

j

ja

0

j

jj G

Ra

n a efor

a e

LetR c

†j

j

j Rj

G

nc

n †e Ree

G

n

n 1

GG

n nn nThm 3.8:

1. 2.

2Gn n

Proof 2:

Ree n e

From 1:

Take trace

1, , Cn

2Gn n

Example: C2 = { e, a }

1 0

0 1

e a

a e i i

g g jj

e eae aaR

a a aae aa

D a

e eee eaR

e a aee ea

D e

0 1

1 0

2Re

0Ra

1 2R

C2 e a

1 1 1

2 1 –1

R 2 0

R can be diagonalized by a similarity transform

1 1

1 1S

1g gS S

with so that

1 0

0 1e

1 0

0 1a

i ia a jj

j

ia

1

0i jg a g

ifotherwise

i jg a g 1i jg g a

DR(a) can be obtained from the multiplication table gi vs gj1

by setting all entries that equal to a to 1, and all others to 0.

Calculation of DR

Example : C3v = { E, C3, C32, σ1, σ2, σ3 } e C3

2 C3 1 2 3

C3 e C32 3 1 2

C32 C3 e 2 3 1

1 3 2 e C3 C32

2 1 3 C32 e C3

3 2 1 C3 C32 e

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

RD e

3

0 0 1 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 1 0 0

RD C

23

0 1 0 0 0 0

0 0 1 0 0 0

1 0 0 0 0 0

0 0 0 0 0 1

0 0 0 1 0 0

0 0 0 0 1 0

RD C

1

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

RD

2

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 1 0 0

0 0 1 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

RD

3

0 0 0 0 0 1

0 0 0 1 0 0

0 0 0 0 1 0

0 1 0 0 0 0

0 0 1 0 0 0

1 0 0 0 0 0

RD

C3v e 2C3 3 v

A1 1 1 1

A2 1 1 –1

E 2 –1 0

R 6 0 0 A1+A2+2E

Comments:

For any finite group G, its DR is equivalent to the block diagonal form

1

2

2

2

1

C

C

C

n

n

n

D

D O

D

D

O

D

n Blocks n Blocks

See App III

3.8. Direct Product Rep, Clebsch- Gordan Coefficients

Definition 3.8: Direct Product Vector Space U V

Let U & V be inner product spaces with orthonormal bases

1, ,i Ui nu 1, ,j Vj nv & respectively

The direct product space of U & V is defined as the inner product space

i j i j i j i jk l k l k l k lU V

w w u v u u u u v v

W U V 1, , ; 1, ,i j i j U Vspan i n j n w u v

with

Thus i ji jx w x W x

† i ji jx y x y *i j i j

i j

x y

Example: 2-particle system

1 2,x x x1 2H H H

3j j j j jd x x x

j jH

1-particle states:

2-particle system:

3 31 2 1 2 1 2d x d x x x x x H

3j j j jd x x x

3 31 2 1 2 1 2,d x d x x x x x 6d x x x

Def: Direct Product Operators

Let U & V be the operator spaces on U & V, resp.

Then the operator space on UV is the set

,U V U VD A B A B L L L

with i j i jD w A B u v i jA u B v

k lk l i jw D k l

k i l ju A v B k lk l i ju v A B

k l k l k li j i j i jD A B A B

Definition 3.9: Direct Product Representation

Let D(G) & D(G) be reps of G on U & V, resp. Then

D G D G D G is a rep of G on U V

k Tr D g

i j

i jD g

k k

kg

i j

i jD g D g

Let D(G) be IRs of G. Then

D a D

with †a †kk

k kk G

n

n

Note: D(G) can be reducible even if D(G) & D(G) are IRs.

Example: S3 S3 e 2 (123) 3 (23)

1 1 1 1

2 1 1 –1

3 2 –1 0

11 1 1 1 1

12 1 1 –1 2

13 2 –1 0 3

22 1 1 1 1

23 2 –1 0 3

33 4 1 0

1

14 2 1 3 0

6a 1

2

14 2 1 3 0

6a 1

3

18 2 1 3 0

6a 1

3 3 1 2 3

Comments:

D a D

1 1

Cn a

W U V W

are invariant subspacesW

For a properly chosen basis, D is block diagonal.

E.g.

1, , ; 1, , ; 1, ,k Cw k n a n

with the order of varying indices being: k

Definition 3.10: Clebsch-Gordan Coefficients ( CGC )

i jk i j kw w w w

, , i jki j k w w

[ wi j = uivj complete ]

, ,k i jw w i j k

( sum over i, j )

( sum over i, j )

Theorem 3.12: Orthonormality & Completeness of CGCs

, , , , i ji ji j k k i j

, , , , kkk i j i j k

*, , , ,k i j i j k where

Proof:

, , i jki j k w w

Follows directly from the orthonormality & completeness of wi j since

sum: ,,k

sum: i, j

Comments: , ,k i jw w i j k

, ,i j kw w k i j

' '

' '

i j

i j i j i jU g w w D g D g

k

k k kU g w w D g

Setkk w

, i ji j w

, , ,i j k i j k

, ,U g i j U g k k i j

,k

kk D g k i j

, , ,k

ki j i j k D g k i j

, ,i j k

i j kD g D g i j k D g k i j

Theorem 3.13: Reduction of Product Representation

, ,i j k

i j kD g D g i j k D g k i j

, ,k i j

k i jD g k i j D g D g i j k

Applications:

• Broken symmetry

• Addition of angular momenta ~ Wigner-Eckert theorem ( Chap 7 )

Short-cut to Theorem 3.13: D a DD

,

, , , ,i j D g D g i j Di j g i j

, , , ,k

kk i j i j D g D g i j i j k D g

, , ,

, ,i j

i jk k

D gD g D g i j k k k k i j

, ,k

kk k

i j k D g k i j

, ,i j k

i j kk i j D g D g i j k D g

,

k D Dg D g k k g k