PHY 745 Group Theory
11-11:50 AM MWF Olin 102
Plan for Lecture 4:
The “great” orthogonality theorem
Reading: Chapter 2 in DDJ
1. Schur’s lemmas
2. Prove the “Great Orthgonality
Theorem”
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PHY 745 Spring 2017 -- Lecture 4
1
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PHY 745 Spring 2017 -- Lecture 4
2
The great orthogonality theorem on unitary irreducible
representations
Notation: h  order of the group
R  element of the group
 ( R )  ith representation of R
i

denote matrix indices
li  dimension of the representation
   ( R)    ( R)
i
R
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*
j
h
  ij  
li
PHY 745 Spring 2017 -- Lecture 4
3
Proof of the great orthogonality theorem
• Prove that all representations can be unitary
matrices
• Prove Schur’s lemma part 1 – any matrix which
commutes with all matrices of an irreducible
representation must be a constant matrix
• Prove Schur’s lemma part 2
• Put all parts together
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PHY 745 Spring 2017 -- Lecture 4
4
Proof of the great orthogonality theorem
• Prove that all representations can be unitary
matrices
• Prove Schur’s lemma part 1 – any matrix which
commutes with all matrices of an irreducible
representation must be a constant matrix
• Prove Schur’s lemma part 2
• Put all parts together
Schur’s lemma part 1:
A matrix M which commutes with all of the matrices of
an irreducible representation must be a constant
matrix: M=sI where s is a scalar constant and I is the
identity matrix.
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PHY 745 Spring 2017 -- Lecture 4
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Schur’s lemmas part 1:
A matrix M which commutes with all of the matrices of
an irreducible representation must be a constant
matrix: M=sI where s is a scalar constant and I is the
identity matrix.
Proof:
Suppose M is a diagonal matrix d , the premise becomes
d  i ( R )   i ( R) d for all elements of the group R
  d i ( R)    i ( R)d 
kl
kl
d kk  i ( R )    i ( R)  dll
kl
kl
 d kk  dll   i ( R) kl  0
 d kk  dll
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for all k , l and for all R
if i ( R ) is irreducible
PHY 745 Spring 2017 -- Lecture 4
6
Example irreducible representation:
1 0
3
 E  

0
1


1


3  C    2
 3
 2
  12
3
  B  
 3
 2
1 0 
3
  A  

0

1



3
2
1
2
 3
  12
   D  

 3

 2
 3
  12
  F   
1
 3
2
 2
3
2
3
2
1
2






1 
2 

3
2
Extension of proof to more general matrix M :
Note that M  H1  iH 2
where H1  M  M †
H 2  i  M  M † 
In these terms, the premise is:
 H1  iH 2  i ( R)  i ( R)  H1  iH 2  for all elements of the group R
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PHY 745 Spring 2017 -- Lecture 4
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 H1  iH 2  i ( R)  i ( R)  H1  iH 2  for all elements of the group R
 H1 i ( R)   i ( R) H1
and
H 2 i ( R )   i ( R ) H 2
Hermitian matrices can be diagonalized by a unitary
transformation U
UH1i ( R )U †  U  i ( R ) H1U †
for all R
UH1U †U i ( R )U †  U  i ( R )U †UH1U †
d  'i ( R )   'i ( R) d
where  'i ( R )
for all R
is an equivalent representation of the group
The same construction holds for H 2 ,
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PHY 745 Spring 2017 -- Lecture 4
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Proof of the great orthogonality theorem
• Prove that all representations can be unitary
matrices
• Prove Schur’s lemma part 1 – any matrix which
commutes with all matrices of an irreducible
representation must be a constant matrix
• Prove Schur’s lemma part 2
• Put all parts together
Schur's lemma part 2
Consider two irreducible representations of the same group
1 ( R ) with dimension
1
and  2 ( R) with dimensions
Suppose there exists a rectangular
M 1 ( R )   2 ( R ) M for all R.
1

2
2
.
matrix M such that
It follows that either M  0
or 1 ( R ) and  2 ( R) are equivalent.
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PHY 745 Spring 2017 -- Lecture 4
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Schur's lemma part 2
Consider two irreducible representations of the same group
1 ( R ) with dimension
1
and  2 ( R) with dimensions
Suppose there exists a rectangular
M 1 ( R )   2 ( R ) M for all R.
1

2
2
.
matrix M such that
It follows that either M  0
or 1 ( R ) and  2 ( R) are equivalent.
Suppose M 1 ( R )   2 ( R) M
for all R
 M  ( R)     ( R) M 
  ( R)  M  M   ( R) 
†
1
1
†
†
2
†
†
2
†
1 ( R 1 ) M † = M † 2 ( R 1 )
 1 ( R ) M † = M † 2 ( R)
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for all R
for all R
PHY 745 Spring 2017 -- Lecture 4
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Schur’s lemma part 2 -- continued
M 1 ( R )   2 ( R) M
for all R
1 ( R ) M † = M † 2 ( R )
for all R
M 1 ( R ) M †  MM † 2 ( R )   2 ( R ) MM † for all R
M † 2 ( R ) M  M † M 1 ( R )  1 ( R ) M † M for all R
1
M
M 
2
†
2
1
1
M M
†
MM 
†
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1
2
2
PHY 745 Spring 2017 -- Lecture 4
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Schur’s lemma part 2 -- continued
MM † 2 ( R )   2 ( R ) MM † for all R
M † M 1 ( R )  1 ( R ) M † M for all R
From Schur's first lemma:
Consider the case where
1

2
MM †  s2 I
(
M † M  s1I
( 1  1)
1


2
)
, it follows that s1  s2  0
otherwise there is an inconsistency  MM
Consider the case where
2
2
†

†
 M †M
, it follows that s1  s2 *
For the case that s1  0 :
M 1 ( R )   2 ( R) M
 1 ( R)  M 1 2 ( R ) M for all R
 Representations are equivalent
For the case that s1  0 :
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MM † PHY
 0745 Spring
 2017
M
0
-- Lecture 4
12
Proof of the great orthogonality theorem
• Prove that all representations can be unitary
matrices
• Prove Schur’s lemma part 1 – any matrix which
commutes with all matrices of an irreducible
representation must be a constant matrix
• Prove Schur’s lemma part 2
• Put all parts together
   ( R)    ( R)
i
R
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*
j
h
  ij  
li
PHY 745 Spring 2017 -- Lecture 4
13
Construct a
2

1
matrix
M    2 ( R ) X 1 ( R 1 )
R
Note that:  2 ( S ) M  M 1 ( S )
where S is a member of the group
Details:  2 ( S ) M    2 ( S ) 2 ( R) X 1 ( R 1 )
R
   2 ( SR ) X 1 ( R 1 )
R
   2 ( SR ) X 1 ( R 1 )1 ( S 1 )1 ( S )
R
   ( SR ) X  ( SR  )1 ( S )
2
1
1
R
=M 1 ( S )
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PHY 745 Spring 2017 -- Lecture 4
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Proof continued:
   ( R)    ( R)
*
i
j
R
Construct a
2

1
h
  ij  
li
matrix
M    2 ( R ) X 1 ( R 1 )
R
Since:  2 ( S ) M  M 1 ( S )
for
2

1
where S is a member of the group,
M  0 for arbitrary X . Choosing particular indices:

2
1
1 
2
1
1

(
R
)
X

(
R
)


(
R
)

(
R
)

 




R
 R

    ( R) 
2
R
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
  ( R) 
1
PHY 745 Spring 2017 -- Lecture 4
*

0
15
Proof continued:
Construct a
2

   ( R)    ( R)
*
i
j
R
1
matrix
h
  ij  
li
M    2 ( R ) X 1 ( R 1 )
R
Since:  2 ( S ) M  M 1 ( S )
where S is a member of the group,
for 2  1 M  sI for arbitrary X . Choosing particular indices:

1
1
1 
   ( R ) X  ( R )   s
 R

For particular choice of X :
∗
1
1
1

(
R
)

(
R
)  s   s 1
 

R

1
1
1

(
RR
)

h

 
 ( E )  h   s 1 
R
s
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h
1
PHY 745 Spring 2017 -- Lecture 4
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