PHY 745 Group Theory
11-11:50 AM MWF Olin 102
Plan for Lecture 2:
Representation Theory
Reading: Chapter 2 in DDJ
1. Review of group definitions
2. Theory of representations
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Group theory
An abstract algebraic construction in mathematics
Definition of a group:
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Some definitions:
Order of the group  number of elements (members) in
the group (positive integer for finite
group, ∞ for infinite group)
Subgroup
 collection of elements within a group
which by themselves form a group
Coset
 Given a subgroup gi of a group a
right coset can be formed by multiply
an element of g with each element of gi
Class
members of a group generated by the
conjugate construction C = X i1YX i where
Y is a fixed group element and X i are all of the
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elements of the group.
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Example of a 6-member group E,A,B,C,D,F,G
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Subgroups:
E
(E , A)
( E , D, F )
( E , B)
(E , C )
Classes:
C1  E
C2  A, B, C
C3  D, F
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Example of cyclic group of order 4:
E
A
A2
A3
E
E
A
A2
A3
A
A
A2
A3
E
A2
A2
A3
E
A
A3
A3
E
A
A2
Subgroups:
E
(E , A2 )
Classes:
C1  E
C2 =A
C3 =A2
C4 =A3
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Example of non-cyclic group of order 4
E
A
B
C
E
E
A
B
C
A
A
E
C
B
B
B
C
E
A
C
C
B
A
E
Subgroups:
Classes:
E
C1  E
C2 =A
C3 =B
C4  C
(E , A)
(E , B )
( E, C )
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Representations of a group
A representation of a group is a set of matrices (one
for each group element) -- ( A), ( B)... that satisfies
the multiplication table of the group. The dimension
of the matrices is called the dimension of the
representation.
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Example:
Note that the one-dimensional "identical representation"
1 ( A)  1 ( B)  1 (C )  1 ( D)  1 ( E )  1 ( F )  1 is always possible
Another one-dimensional representation is
 2 ( A)   2 ( B )   2 (C )  1
 2 ( E )   2 ( D)   2 ( F )  1
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Example:
A two-dimensional representation is
1 0
 E  

0
1


1 0 
  A  

0

1


3
1


3  B    2
 3
 2
1


3  D    2
 3
 2
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3
2
1
2
 3
  12
  C  

 3

 2
 3
  12
  F   
1
 3
2
 2
3
2

3
2
1
2






1 
2 

11
3
2
What about 3 or 4 dimensional representations for this group?
For example, the following 3 dimensional representation satisfies
the multiplication table:
1 0 0
  E    0 1 0 
0 0 1


1 0

  C    0  12

3
0

2

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1 0 0 
  A    0 1 0 
 0 0 1


0 
1 0


3
 2    D    0  12

3
1 
0

2 
2

1 0 0 


3
1
 B   0  2 2 

3
1 
0
2
2 

0 
0 
1 0



3
3
1
F   0  2  2 
2 


3
1 
 12 
0

2
2 

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Reduced form of representation
1 0 0
  E    0 1 0 
0 0 1


1 0

  C    0  12

3
0

2

1 0 0 
  A    0 1 0 
 0 0 1


0 
1 0


3
 2    D    0  12

3
1 
0

2 
2

1 0 0 


3
1
 B   0  2 2 

3
1 
0
2
2 

0 
0 
1 0



3
3
1
F   0  2  2 
2 


3
1 
 12 
0

2
2 

In this example, the “irreducible” representations for the
group are 2 one-dimensional and 1 two-dimensional
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Comment about representation matrices
A representation is not fundamentally altered by
a similarity transformation
( A)  S 1( A) S
Check: Suppose ( AB )  ( A)( B)
( AB )  S 1( AB ) S  S 1( A)( B ) S
=S 1( A) SS 1( B ) S
=( A)( B)
• Typically, unitary matrices are chosen for
†
†
unitary
matrix:
UU

U
U 1
representations
• Typically representations are reduced to block
diagonal form and the irreducible blocks are
considered in the representation theory
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y ... z 
 x
 y * w ... u 

( A)  

 similarity transformation


 z * u * ... w 






1
block diagonal form
( A)
0
0
0
 2 ( A)
0
0
0
0
0
0


0 
0 

n
 ( A) 
0
distinct
irreducible
representation
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The great orthogonality theorem
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
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Great orthogonality theorem continued
   ( R)    ( R)
*
i
R
j
h
  ij  
li
Analysis shows that
l
2
i
h
i
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Simplified analysis in terms of the “characters” of the
representations
lj
 j ( R )   j ( R) 
 1
Character orthogonality theorem
   ( R)  
i
*
j
( R)  h ij
R
Note that all members of a class have the same
character for any given representation i.
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What the great (“wonderful”) orthogonality theorem will do
for us:
1. Show that there are a fixed number of distinct
irreducible representations and help us find them
2. Show that the irreducible representations of a group
have properties of orthogonal vector spaces.
3. Result in a simplified orthogonality theorem based on
the “characters” of the group
Often the irreducible representations are related to physical
quantities such as quantum mechanical wavefunctions or
operators.
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Example similarity transformation
Let
 cos
M 
 sin 

1
1
S MS  S 


1
2
i
2
e  i
e  i
 sin  
cos  

S 


ei   e  i

 i i 
e   0
2
1
2
1
2
i
2


i 
2 
1
2

1
S 


1
2
1
2


i 
2 
i
2
0
i 
e 
Note: Two representations i ( R ) and i '( R ) are equivalent
if i '( R )  S 1 i ( R ) S
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