3
REPRESENTATION THEORY
OF
FINITE GROUPS
of
In the
first
gToups
notions
we discussed some elementary In Chapter 2, we
sets.
chapter,
and
concepts of elements
in terms of the abstract
and studied their proper
spaces
acting on their Hilbert
operators
treated
In physics, we are interested
to quantum physics.
relevant
are
ties which
spaces of physi
act on suit able Hilbert
which
transformations
a 'state' of the
in groups of
the Hilbert space characterizing
of
vector
each
cal systems,
matrix repre
we have introduced the concept of
system. In Section 2.2,
natural to combine
It is therefore
in a Hilbert space.
senting an operator
all the elements of
to obtain matrices representing
these two concepts and
theory
under the representation
a group. The study of such matrices comes
although
shall consider finite groups only,
of groups. In this chapter, we
modified
easily
or
can
be
they
are
most of the results either hold good as
a
groups
the case of infinite groups. Continuous
in
the
next
chapter.
are dealt with
and their
representations
to
3.1 INTRODUCTION
=
Definition. Let G
{E, A, B, C,...}be a finite group of order
gwith E as the identity element. Let T
{T(E), T(A), T(B), ...} be a
collection of nonsingular
square matrices, all of the same order,having the
=
property
T(A)T(B)
That is, if AB
= T(AB).
(3.la)
=C
in the group G, then
= T(C),
T(A)T(B)
then the collection T
G. The order of the
of
matrices is said to be a representation
of the
matrices of
T
is
called
(3.1b)
g
the dinension of the represe"
tation.
Let
act,
Ln be an n-dimensional
of G
space on which the operators
Let {o}be an
orthonormal basis in Ln. The operation of an element
vector
Representation
Theoryof Finite Gronps
53
AEG on a basis vector is then given by (see (2.19) and (2.20)1
(3.2)
where 7(A) is the
(.).
matrix representing
A with the basis
An elerment
then be, in analogy with
(2.21), given by
atthe matrix T(4)could
Tj(A) =(o,, Abi).
1We could similarly obtain
matrices
(with the same basis
a representation
corresponding
(o}). Tt then obvious
of G, for, on the one hand,
ABO,
while, on theother
is
= A•,T(D) =
(3.3)
to all the elernents of
G
that these mattices generate
Tiy(AYT,(D),
hand,
ABo,=
Since the above two operations
oTi(AB).
must give the same result, we have
T,(A)T (B) = Ti(AB) visi, ksn;
j=l
T(A)T(B)
which
is just
=T(AB),
(3.la).
that T is a group under
However, one must be careful here because the
of T need not be distinct. If each distinct matrix of T is taken
One may be tempted to jump to the conclusion
matrix
multiplication.
matrices
only once, the resulting set is certainly a group under matrix multipication.
Hereafter, whenever we refer to the 'group' T, we shall
of the distinct matrices of T.
really
mean the set
If all the matrices of T are distinct, there is clearly a gne- toone corre
spondence between the elements of G and the matrices of T. Ia this case,
the groups G and T are isomorphic to each other and the repzesentat ion
generated by the matrices of T iscalled a faithful representation of G Ou
Lhe other hand,if the matrices of T are not distinct, there exists oaly a
Jomomorphism fron G to T and such a reptesent ation is calleda unfaithful
representation of G.
The simplest representation of a group is obtained when we associate
unity
with every elenent of the group. Thus,in our exanple of the group
Ca (ef. Section 1.1.2), we would have the corespondence
Elenent
Representation
A
constant
auunber is
1
a
special
1
case of a
1
matrix-itis a squase natix of order oue.
54
Elements of Group
The
set
general.
(1,1,...,
1) does indeed
Theory for Physicists
of
form a representation
say,
of two elerments,
For example, the
product
the above case,
correspondsto l x
This is known as the
identity
The identity
representation
any group
= u
C4 m,
in
1=l the considered representation.
in
representation.
of
any
is clearly an unfaithful representation
reprea faithful
matrices of Problem 1.1(v) is
to Cavshown in Problem 1.6, it isisomorphic
group. The set of the eight
sentation of Cav, because,as
Every group has at least one
left to Problem 3.14.
faithful representation,
the proof
of
which is
of a group. We note
AE A
element E of G has the property that EA
G. In terms.of the matrices of a representaton,this
3I.2 Some properties of representations
that the identity
for all elements
implies that
in
=
=
AE
T(E)T(A)
=T(A)T(E) = T(A)
(3.4)
We see that this matrix equation is satisfied only if
identity
T(E)
= E, the unit
element of the group
matrix. Thus, in any represent ation, the
order.
must be represented by the unit matrix of the appropriate
On taking A-1 for B in (3.la), we see that
=T(AA-l) =T(E) = E,
T(A-') = T(A)".
T(A)T(A-l)
Or
(3.5)
of an element is
This is to say that the matrix representing the inverse
the element.
equal to the inverseof the matrix representing
Suppose we have two representationsof a group G given by
o
If
T;
= {Ti(E), T1(A),...}
and
there exists a nonsingular matrix
= S-T,(A)S,
S such that
Ti(B)
Ti(A)
for all the elements of the group
T;= (T;(E), T;(A),...}.
=S-T(B)S,
b
etc.,
(3.6)
G, then T and T, are said to be eguivalent
representationsof G. This means that the matrices of the first set can be
obtained from those of the second set by a similarity
coordinate vectors of the vector space in which
are defined.
transformation of the
both the representations
We express this by writing in short
T =S-T,S.
(3.7)
two representationsof a group are not equivalent to each other, they are
said to be incquivalentor distinct represent ations.
If
ZIn accordance
identity operator
with our convention,
and the unit matrix.
we shall use the same symbolE
to denote the
55
Theory of Finite Groups
Representation
3.2 INVARIANT
SUDSPACES AND
REDUCIBLE REPRESENTATIONS
It is evident that the vector space L, which is used to generate a rep
of the group G has the following property: For every elernent
resentation
and every vector
E Ln, A¢ also belongs to L,. We say that the
space L, is closed under the transformations of G or, simply closed
under G. It means that the operation of any elerment of G on any vector
A
of
G
vector
of L, does not take us outside
Ln.
A vector space Lm is said to be a subspaceof another vector spaceL,
every vector of Lm is also contained in L,. Lm is called a proper subspace
of L, if the vectors of Lm do not exhaust the space Ln. Thus L,
also
a
if
is
subspace of itself
The vector
but,of course, not proper.
space Ln, which is closed under G, may
possess a proper
subspace Lm which is also invariant under G. In such a case, Lm is said
to be an invariant subspace of LIn under G, and the space L, is said to be
reducible under G.
Reducibility of a representation. Let, as before, {T(E),T(A),
if L, has
T(B),...}be a representation of G in Ln. We now state that
G, then in a suitable basis the
<
n)
under
Lm
(m
an invariant subspace
have the form
matrices of the representation
3.2.1
=pora
(3.8)
T(4)
order
m and n- m
D)(4) and D(A) are square matrices isofa null matrix of order
(n -m) x m and 0
respectively, X(A) is of order
where
m x (n-m). To show this,we use the row vector notation
;=
(000 ... 1;
0
for the vectors:
(3.9)
...0),
the other elements
column has unity and
which means that the i-th
be chosen
the n basis vectors may conveniently
zero. The labeling of
all
are
in
such a way that the first
AEG on a basis vector ,,(1<u < m)
a
is
then given by
:
Ti
... Tim
Tm1
..Tmm :Im,m+l
li,m+1
=
Ad, (000... 1, 0 ... 0)
sd
.
m basis vectors are in Lm: The operation
of
Tin
Imn
..Tn+1,n
Tn+1,1
...Thm
i
Th,mtl
.. Thn
(3.10)
Ta,mtl
= (TiT2 ...Tum
Tun),
Elements of Goup Theory
56
for Physicists
8ake of brevity. Now, since
Ti, for Tii(A) for the
also belongs
vector
transformed
A,
where we have written
the
Lm is itself invariant under G,
to Lmi hence
components along
the basis vectors,
its
m+1, m+2, ...,o,
must be zero, i.e.,
m+ 1<k<n.
T,(4) = 0,
(3.11)
all the
1 to m, we see that
arbitrary, and letting it run from
top
right
at
the
x
of order
elements in the rectangular block
in
(3.8).
shown
Hence T(A) has the form
corner of T(4) must be zero.
C.
of
two elements of the group G, say,
the
product
Let us consider
is
However,
(n– m)
m
AB =
In
T(A)T(B)
= T(C),
or
D()(A)
T(C)
above, we have
considered
of the representation
terms of the matrices
=
D)(B)o
0
:
:
D?(A)J
X(B)
D')(A)D)(B)
:
X(A)
D)(B) J
D)(A)D(B) J(3.12)
Lx(4)p(B) +D?(A)X(B)
But T(C) must itself be of the form
0
=
T(C)
D(C)J
L X(C)
Therefore,
we have
D(A)D)(B)=D(C), D(A)D(B) =D)(C),
(3.13)
and
X(A)D
From (3.13), it is clear
(B)
+D®(A)X(B) =X(C).
(3.14)
sets of matrices
D(1) = {D(E),
D(A),...)and Da) =
{D(2)(E),
that the two
D(2(A),...) also give us two
representations
of
new
dimensions
and
It is also cdear
respectively for the
that the basis
group
G.
vectors
{fo1,2,...,o) are the basis
representation
for the
and the remaining n
for D(?)
basis vectors
n-m
m
D)
In this case, T
m
is 6aid
that the reducibility
to be
{m+l,:.On)
a reducible
representation.
Thus, we see
representation is
çonnected
with the
invariant subspace of
eristence oJ
the full space.
shall denote the
of a
a proper
We
n- =p-dimensional vector
m
basis vectors
(m+l**n}by Lp:
space defined by
the
Representation Theory of
3.2.2
Finite
57
We shall now show that any
T of finite representations.
group,whose matrices
represntation
may
equivalent (through a
be non-unitary,
to a representation
similarity
transformation)
tarv sevices. For this
purOse we
H
theorem from
by a
matrix algebra that a
unitary
matrix,
=AEGTAYr(A),
where the summation is over all the elements
transformation.
If
U
is
(3.15)
of the group
hermitian
is
by uni
a hermitian
define
nalized
Groups
A theorem on
G,We invoke a
matrix can be fully diago
the necessary
transfor rnation,
then
U-HU= Ha,
(3.16)
where Ha is a diagonal matrix whose diagonal
elerments are the (real) eigen
H.Using (3.15)
values of
H
in (3.16),
we have
= U-lT(A)T(A)U
AEG
30u-T(A)UU-T
(A)U
AEG
=AEG T(A)T(4),
=U-T(A)U.
(3.17)
Taking the k-th diagonal
where T'(A)
element of (3.17), we
get
saaie s (Halu=de =T,(A)T!(A)
AG
j
=
Hhoos To inotece supido
ottaiio
T,(A)TI; (A)
AEG j
=
(3.18)
T,(A)°.
>0. But
is non-negative, we have dz
each term in this summation
for all
j
and
0 for all values of
and only if T!.(A)
dt can be zero if
determinant for all
This would give a vanishing
the elemnents A E G.
Hence
excluded.
a case which we have
representation,
=
Since
the
>0, that is, must be
the
d
matrices
of
d
a nonsingularmatrix. We
consequence,
Ha simply by taking the
any power of the matrix
As a
it
is
positive.
also clear
that Ha
is
can therefore obtain
diagonal elemets of H,
corresponding power of all the
ie.,
(Ha)] =(d)',
A
matrix
all
of
are
whose eigenvalues
positive is called
(3.19)
a po sitive
definite
matri.
Elements of Group Theory for Physicists
58
negative.
where p is any real number,positive or
matrix which converts the non.
is then seen to be
matrices
The required similarity transformation
unitary matrices
T(A) into unitary
(A)
V= UH,
(3.20)
giving
r(A) = V-lT(A)V
(3.21)
=Ha 2u-T(A)U HY?
= H;T(A)H".
To verify that the matrices
(3.22)
(A)are indeed unitary, we note that
=[H;T(A)H}H}T(A)H,)
=HT(A)H4T(A)H,/2
r(4)r'(4)
= H,T(A)
T(B)T"(B)T"(A) H,/2
by (1.9)
BEG
=H
BEG
T(AB)T" (AB)H, /2
= H,H H,/2 by
(3.17)
= E,
which shows that r(A) is a unitary
matrix.
G are unitary operators
the elements of the group
formation of the representation
If
physical
the
trans
T to the representation similarity
r has a simple
meaning-it implies going over from an
oblique
nate axes to an orthonormal
one. The
system of coordi
nonunitary nature of the matrices
T(A), etc., indicates that the basis
vectors of
the representation T,
are not
Ln,chosen as the basis for
orthonormal, whereas the representation 1
by unitary matrices shows
that the basis vectors for
the
are orthornormal.
We have achieved
coordinate system, say
=
representation
this transformation
1
from the oblique
(1,2,.. .,o,),to the orthonormal
coordinate
U2,... ,n), by means of
the matrix
of
that
(3.20) so
V. In this light, what
we have said in this theorem is
very simple and almost
really
trivial:
It is possible to
choose an orthonormal set
of basis vectors in any
finite dimensional
vector space, which is
true! The difficulty
obviousiy
in extending this
theorem to
resentations
infinite-dimensional
rep
or to the
representations
of infinite groups
is regarding
COnvergence
of the
various sums
encountered in its proof. The
may none the less be
theoreit
proved to hold for cert
ain classes of
known as compact groups
infinite
group
which will be treated
in the next
Owing to this theorem,
chapter.
hereafter, we need to
by
consider representau
unitary matrices only.
This no doubt affor
ds a great
simplification.
system, say
y=
= (U1,
V
59
Theory of Finite Groups
Representalion
a Ireduciblerepresentations. If the representation
above is reducible, the representation
=
T consid
{r(E), r(A), .. .}, defined
by (3.21), is also reducible, since they are defined in the same space and
are equivalent, Moreover, since the matrices of Ir are unitary, they must
have the form
ered
r(A)
=
sO(A)
:
(3.23)
etc.,
where we have the two representations
by unitary
matrices
=
S()
= {s0(E),
s)(A),:..) and S()
(sO(E), S)A),...) which are defined in the
spaces Lm and Lp and hence are equivalent to D) and D(2) respectively.
and S(2) are further
It may be possible that the representations
S)
reducible, i.e., the spacesLm and L, may contain furtherinvariant (proper)
subspaces within them. This process can be carried on until we can find no
Initary transormation
which reduces all the matrices of a representation
further. Thus, the final form of the matrices of the representation T may
look like
rO(A)
r(4)
rO(A)
=
etc.,
bL
0srearoteso
with
all
the matrices
of T
a complete reduction
(3.24)
ro(A) J
havingthe same redu ced structure. When such
of a representation is achieved, the component repre
r(), r2),...
are called the irreducible representations
of
is said to be fully redu ced.
the group G and the representation
It may be noted that an irreducible representation
may occur more than
once in the reduction of a reducible representation T. The matrices of the
representation T,are just the direct sum of the matrices of the component
irreducible representations and this may be denoted by
,r)
sentations
aaie aI=a1r
r
azr
a,
r()
=a;
r4),
where, in the last step, the symbol for summation
(3.25)
is
to be understood
in
the sense of direct sum.
may appear from (3.24) that the number of distinct
irreducible representations of a group is very large and unlimited. However,
At first
sight,
it
60
for Physicists
Elements of Group Theory
the irreducible
is not the case,because
this
groups,
representations
for finite
which limit their number and
conditions
of a group satisfy various
which
of the
aPplications
in
the
useful
are, at the same time, very
few sections we take up the
the next few
In
problems.
groups to physical
examn study
As
an
representations.
irreducible
of
the
of such properties
C4y are discussed in Section 3.6
irreducible representations of
renres.
use the abbreviation IR for irreducible representa
We shall henceforth
thch
tions'.