1. 3. Group Representations
3.2 Irreducible, Inequivalent Representations
3.3 Unitary Representations
3.4 Schur's Lemmas
3.5 Orthonormality and Completeness Relations of Irreducible Representation Matrices
3.6 Orthonormality and Completeness Relations of Irreducible Characters
3.7 The Regular Representation
3.8 Direct Product Representations, Clebsch-Gordan Coefficients

2. 3.1. Representations Def: Group of Operators / Linear Transformations
= Set of invertible linear operators on a linear space that is also a group
Definition 3.1: Group Representations
Let L be the space of linear operators on an n-D vector space V.
Let U: G  L, g  U(g) be a homomorphism, i.e,
U(g g') = U(g) U(g')  g, g'  G
Then U(G) = { U(g) | gG } is a n-D representation (rep) of G.
The rep is faithful if U is an isomorphism.

3. 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
(Einstein's summation notation) 
 D(G) forms a matrix representation of G.
Example 1: Trivial 1-D Rep for Every Group, V = C
U: g  U(g) = 1  gG

4. 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

5. Example 5: R(2) = { R(), 0   < 2 } , V = E2

6. Similarly 
Also
{ U() , 0   < 2 } is a 2-D representation of R(2) on E2
{ D() , 0   < 2 } is a 2-D matrix rep of R(2) wrt basis {e1, e2 }
Example 6: D3 or C3v = { E, C3, C32, σ1, σ2, σ3 } , V = E2
The 2-D matrix rep is

7. Example 7: Vf = { f(x,y) = a x + b y | x,y R, a,b  C }
Vf 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 E2 so that
 x = {x,y}  E2 & g G
This induces a rep of G on Vf as follows
where 
Let g = h k, then
 This mapping is indeed a homomorphism
The ensuring 2-D matrix rep's are identical to those found earlier ( prove it ! )
Vf is an (scalar) example of a field.
Higher dim versions  vector, spin, isospin, …

8. Let U be a rep of K on V so that
Theorem 3.1: G/H
If G has a non–trivial invariant subgroup H, then any rep of K = G/H is also a degenerate rep of G.
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:
Let U be a rep of K on V so that
U: K  L, k  U(k) L = space of linear operators on V.
: G  L, g  (g) = U(k) where g h = k  K for some hH
is a homomorphism.
  is a rep for G.
Mapping is many–1 since H is non-trivial.
Follows from Theorem 2.5
Theorem 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'

9. Example: S3 = { e, (123), (132), (23), (13), (12) }
Corollary: All rep's, except the trivial one, of simple groups are faithful.
e.g., Cn with n prime
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)

10. 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 gG in rep U(G) is defined as
For a matrix rep D(G),
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  xV1 & gG
V1 is proper / minimal if it doesn't contain any non-trivial invariant subspace

11. 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 = E2 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.

12. Example 2: R(2) V = E c.f. Eg 5, §3.1
1-D invariant subspaces wrt R(2) are spanned by 
Matrix rep wrt { e+, e– } is
Example 3: D3 = { E, C3, C32, C2, C2', C2'' } V = E2
E2 is minimal wrt D3
 2-D rep in Problem 3.1 is an IR

13. 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.  
where D1(g) is n1 n1
Since
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.

14. 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

15.  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.
V1 is invariant 
Orthogonal complement 
U(g) is unitary  
QED

16. 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  gG if S satisfies
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 )
Proof of unitarity of U(G) is as follows:

17. 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)
where U is an IR that occurs n times.
D(G) will be block diagonal for a properly chosen basis

18. 3.4. Schur's Lemmas Schur's Lemma 1:
Let U(G)  L be an IR of G on V, and A  L. Then
AU(g) = U(g)A  gG  A =  E C
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 :
where { |  j  } can be chosen to be orthonormal:

19. Label  is introduced to account for possible degeneracies, viz.,
A U(g) = U(g) A 
is also an eigenvector belonging to j
Let n be the degree of degeneracies.
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

20. 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 pG,
U(p) U(g) = U(g) U(p)  gG
Schur‘s 1st lemma  U(p) = p E  pG
 U(G) is equivalent to the 1-D rep { p  p  C }

21. 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  gG
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' }
 R is an invariant subspace of V wrt U(G)
U(G) is IR 
Either R = { 0 } & hence A = O
or R = V so that map A is onto
Proof is done if for 2), A is also 1-1

22. Let N' = Null ( V' ) = { x'  V' | A x' = 0 }
i.e., N' is an invariant subspace of V' wrt U’(G)
U' (G) is IR 
Either N' = V' & hence A = O
or N' = { 0 } so that map A is 1–1 since
Thus, A  O is an isomorphism &
A U' (g) = U(g) A  U(g) = A U' (g) A–1

23. 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 gG 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

24. Theorem 3.5a : Orthonormality of IR Matrices
dual
is an nG–D orthonormal vector for given ( , k, m )
Proof to be given after some examples

25. Theorem 3.5: Orthonormality of Unitary IR Matrices
dual
is an nG–D orthonormal vector for given ( , k, m )

26. For Abelian groups, all IRs are 1–D:
Example 1: C2 = { e, a } nG = 2 e a C2 e a
1 1
2 –1
Identity rep:
1-D rep orthonormal to d1 :
Example 2: D2 = { e, a = a–1, b = b–1, c = a b } nG = 4 e a b c
Identity rep: D2 e a b c
1 1 2 –1 3
4
Invariant subgroup { e, a }  C2
Factor group D2 / C2  C2
Example 3: Td = { T(n) | n  Z } nG =  c.f. Chap 1
Td Abelian  All IRs are 1–D
Orthonormality:

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

28. Proof of Theorem 3.5a:
Let X be any nn matrix and
Schur's lemmas:
Either    & MX = 0
or  =  & MX = cX E
Let  
Take trace on both sides  
QED

29. Corollary 1:
Proof:
is an nG–D orthonormal vector for given ( , k, m )
Since for a given rep , there are n2 choices of ( k, m ). 
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

30. Theorem 3.6: Completeness of IR Matrices1) 2)
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 to infinite groups requires existence of an invariant measure to replace group sum with an integral
Thm are basis-dependent; character versions of them are not.

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

32. 3.6. Orthonormality and Completeness Relations of Irreducible Characters
Lemma: Sum over a Class
Let U(G) be an IR of G . Then
Proof:
Let
then
Schur's 1st lemma:
Take trace : 
QED

33. Theorem 3.7: Orthonormality & Completeness of 
Proof:   
QED

34. Define the normalized character as
Orthonormality
Completeness
Since j = 1, …, nC , { j } is an nCnC matrix ( character table ).
Corollary: ( for finite groups )
Number of inequiv IRs = Number of classes = nC
Note: Tables of characters are independent of basis.
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

35. Example: S3 = { e, (123), (132), (23), (13), (12) }
3 classes  3 inequiv IRs
Identity rep: 1 = ( 1, 1, 1 )
1–D rep 2 = ( 1, a, b ). Orthonormality 
 2 = ( 1, 1, –1 )
Dim d of 3 :
3 = ( 2, a, b ). Orthonormality 
 3 = ( 2, –1, 0 ) S3 e
2 (123)
3 (23) 1 1 2 –1 3 2

36. Theorem 3.8: 
Proof:
Take trace of   C2 e a 1 1 2 –1  2
Example: C2 = { e = a2, a }
D(G):
 = ( 2, 0 )
D(G)  D'(G):
Prob 3.6

37. Example: Vibration of NH3
C3v e
2C3
3 v A1 1 z A2 –1 Rz E 2
(x,y), (Rx, Ry)
Nxy Nz
Hxy 6
A1+ A2 + 2E Hz 3
A1 + E
NH3 12
3A1 + A2 + 4E
Translation:
A1+ E
Rotation:
A2 + E
Vibration:
2A1+ 2E

38. Example: Electronic states of NH3
N: 1s2 2s22p3
H: 1s
C3v e
2C3
3 v A1 1 z A2 –1 Rz E 2
(x,y), (Rx, Ry) Ns Np 3
A1 + E Hs
NH3 7
3A1 + 2E
Bonds

39. Theorem 3.9: Condition for Irreducibility
U(G) is IR 
i.e.,
Proof:
Let
If U(G) = U(G), then 
Conversely 
so that
for some 
i.e.,
U(G) = U(G)
Character tables of all crystallographic point-groups are given in most texts

40. 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 } :  
Theorem 3.10: Regular Representation
is an nG–D matrix rep of G
Proof: Let a b = c , where a, b, c  G
a b = c 
QED

41. Comments:
Theorem 3.10 is just a version of the Cayley's theorem ( Thm 2.1 ).
with
Cayley:
Reg rep:
with
where Pa is the nn matrix rep of pa.
Alternative proof for Theorem 3.10: 
where

42. Theorem 3.11: Decomposition of the regular rep R1. 2.
Proof 1: 
Let
Thm 3.8:
Proof 2:
From 1:
Take trace 

43. C2 e a
1 1
2 –1
Example: C2 = { e, a } e a R 2 
R can be diagonalized by a similarity transform
with
so that

44. Calculation of DR 
 DR(a) can be obtained from the multiplication table gi vs gj1 by setting all entries that equal to a to 1, and all others to 0.

45. Example : C3v = { E, C3, C32, σ1, σ2, σ3 } A1+A2+2E e C32 C3 1 2 3–1 E 2 R 6
A1+A2+2E

46. Comments:
For any finite group G, its DR is equivalent to the block diagonal form
See App III

47. 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
&
respectively
The direct product space of U & V is defined as the inner product space
with
Thus

48. Example: 2-particle system
1-particle states:
2-particle system:

49. Def: Direct Product Operators
Let LU & LV be the operator spaces on U & V, resp.
Then the operator space on UV is the set
with 
Definition 3.9: Direct Product Representation
Let D(G) & D(G) be reps of G on U & V, resp. Then
is a rep of G on U  V

50. Let D(G) be IRs of G. Then
with
Note: D(G) can be reducible even if D(G) & D(G) are IRs.

51. Example: S3 S3 e
2 (123)
3 (23) 1 1 2 –1 3 2
11 1 1
12 –1 2
13 2 3
22
23
33 4 1

52. Comments: 
are invariant subspaces
For a properly chosen basis, D is block diagonal.
E.g.
with the order of varying indices being: k    
( sum over i, j )
[ wi j = uivj complete ]
Definition 3.10: Clebsch-Gordan Coefficients ( CGC )
( sum over i, j )

53. Theorem 3.12: Orthonormality & Completeness of CGCs
sum: ,,k
sum: i, j
where
Proof:
Follows directly from the orthonormality & completeness of wi j since

54. Comments: 
Set  

55. Theorem 3.13: Reduction of Product Representation
Applications:
Broken symmetry
Addition of angular momenta ~ Wigner-Eckert theorem ( Chap 7 )

56. Short-cut to Theorem 3.13: